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From Wikipedia, the free encyclopedia

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping VW between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

An important special case is when V = W, an endomorphism of V, sometimes the term linear operator refers to this case[1]. In another convention, linear operator allows V and W to differ, while requiring them to be real vector spaces[2]. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not.

A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension);[3] for instance it maps a plane through the origin to a plane, straight line or point. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring.

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Transcription

You now know what a transformation is, so let's introduce a special kind of transformation called a linear transformation. It only makes sense that we have something called a linear transformation because we're studying linear algebra. We already had linear combinations so we might as well have a linear transformation. And a linear transformation, by definition, is a transformation-- which we know is just a function. We could say it's from the set rn to rm -- It might be obvious in the next video why I'm being a little bit particular about that, although they are just arbitrary letters -- where the following two things have to be true. So something is a linear transformation if and only if the following thing is true. Let's say that we have two vectors. Say vector a and let's say vector b, are both members of rn. So they're both in our domain. So then this is a linear transformation if and only if I take the transformation of the sum of our two vectors. If I add them up first, that's equivalent to taking the transformation of each of the vectors and then summing them. That's my first condition for this to be a linear transformation. And the second one is, if I take the transformation of any scaled up version of a vector -- so let me just multiply vector a times some scalar or some real number c . If this is a linear transformation then this should be equal to c times the transformation of a. That seems pretty straightforward. Let's see if we can apply these rules to figure out if some actual transformations are linear or not. So let me define a transformation. Let's say that I have the transformation T. Part of my definition I'm going to tell you, it maps from r2 to r2. So if you give it a 2-tuple, right? Its domain is 2-tuple. So you give it an x1 and an x2 let's say it maps to, so this will be equal to, or it's associated with x1 plus x2. And then let's just say it's 3 times x1 is the second tuple. Or we could have written this more in vector form. This is kind of our tuple form. We could have written it -- and it's good to see all the different notations that you might encounter -- you could write it a transformation of some vector x, where the vector looks like this, x1, x2. Let me put a bracket there. It equals some new vector, x1 plus x2. And then the second component of the new vector would be 3x1. That's a completely legitimate way to express our transformation. And a third way, which I never see, but to me it kind of captures the essence of what a transformation is. It's just a mapping or it's just a function. We could say that the transformation is a mapping from any vector in r2 that looks like this: x1, x2, to-- and I'll do this notation-- a vector that looks like this. x1 plus x2 and then 3x1. All of these statements are equivalent. But our whole point of writing this is to figure out whether T is linearly independent. Sorry, not linearly independent. Whether it's a linear transformation. I was so obsessed with linear independence for so many videos, it's hard to get it out of my brain in this one. Whether it's a linear transformation. So let's test our two conditions. I have them up here. So let's take T of, let's say I have to vectors a and b. They're members of r2. So let me write it. A is equal to a1, a2, and b is equal to b1, b2. Sorry that's not a vector. I have to make sure that those are scalars. These are the components of a vector. And b2. So what is a1 plus b? Sorry, what is vector a plus vector b? Brain's malfunctioning. All right. Well, you just add up their components. This is the definition of vector addition. So it's a1 plus b1. Add up the first components. And the second components is just the sum of each of the vector's second compnents. a2 plus b2. Nothing new here. But what is the transformation of this vector? So the transformation of vector a plus vector b, we could write it like this. That would be the same thing as the transformation of this vector, which is just a1 plus b1 and a2 plus b2. Which we know it equals a vector. It equals this vector. Or what we do is for the first component here, we add up the two components on this side. So the first component here is going to be these two guys added up. So it's a1 plus a2 plus b1 plus b2. And then the second component by our transformation or function definition is just 3 times the first component in our domain, I guess you could say. So it's 3 times the first one. So it's going to be 3 times this first guy. So it's 3a1 plus 3b1. Fair enough. Now what is the transformation individually of a and b? So the transformation of a is equal to the transformation of a -- let me write it this way -- is equal to the transformation of a1 a2 in brackets. That's another way of writing vector a. And what is that equal to? That's our definition of our transformation right up here, so this is going to be equal to the vector a1 plus a2 and then 3 times a1. It just comes straight out of the definition. I essentially just replaced an x with a's. By the same argument, what is the transformation of our vector b? Well, it's just going to be the same thing with the a's replaced by the b's. So the transformation of our vector b is going to be -- b is just b1 b2 -- so it's going to be b1 plus b2. And then the second component in the transformation will be 3 times b1. Now, what is the transformation of vector a plus the transformation of vector b? Well, it's this vector plus that vector. And what is that equal to? Well, this is just pure vector addition so we just add up their components. So it's a1 plus a2 plus b1 plus b2. That's just that component plus that component. The second component is 3a1 and we're going to add it to that second component. So it's 3a1 plus 3b1. Now, we just showed you that if I take the transformations separately of each of the vectors and then add them up, I get the exact same thing as if I took the vectors and added them up first and then took the transformation. So we've met our first criteria. That the transformation of the sum of the vectors is the same thing as the sum of the transformations. Now let's see if this works with a random scalar. So we know what the transformation of a looks like. What does ca look like, first of all? I guess that's a good place to start. c times our vector a is going to be equal to c times a1. And then c times a2. That's our definition of scalar multiplication time's a vector. So what's our transformation -- let me go to a new color. What is our -- let me do a color I haven't used in a long time, white. What is our transformation of ca going to be? Well, that's the same thing as our transformation of ca1, ca2 which is equal to a new vector, where the first term -- let's go to our definition -- is you sum the first and second components. And then the second term is 3 times the first component. So our first term you sum them. So it's going to be ca1 plus ca2. And then our second term is 3 times our first term, so it's 3ca1. Now, what is this equal to? This is the same thing. We can view it as factoring out the c. This the same thing as c times the vector a1 plus a2. And then the second component is 3a1. But this thing right here, we already saw. This is the same thing as the transformation of a. So just like that, you see that the transformation of c times our vector a, for any vector a in r2 -- anything in r2 can be represented this way -- is the same thing as c times the transformation of a. So we've met our second condition, that when you when you -- well I just stated it, so I don't have to restate it. So we meet both conditions, which tells us that this is a linear transformation. And you might be thinking, OK, Sal, fair enough. How do I know that all transformations aren't linear transformations? Show me something that won't work. And here I'll do a very simple example. Let me define my transformation. Well, I'll do it from r2 to r2 just to kind of compare the two. I could have done it from r to r if wanted a simpler example. But I'm going to define my transformation. Let's say, my transformation of the vector x1, x2. Let's say it is equal to x1 squared and then 0, just like that. Let me see if this is a linear transformation. So the first question is, what's my transformation of a vector a? So my transformation of a vector a-- where a is just the same a that I did before-- it would look like this. It would look like a1 squared and then a 0. Now, what would be my transformation if I took c times a? Well, this is the same thing as c times a1 and c times a2. And by our transformation definition -- sorry, the transformation of c times this thing right here, because I'm taking the transformation on both sides. And by our transformation definition this will just be equal to a new vector that would be in our codomain, where the first term is just the first term of our input squared. So it's ca1 squared. And the second term is 0. What is this equal to? Let me switch colors. This is equal to c squared a1 squared and this is equal to 0. Now, if we can assume that c does not equal 0, this would be equal to what? Actually, it doesn't even matter. We don't even have to make that assumption. So this is the same thing. This is equal to c squared times the vector a1 squared 0. Which is equal to what? This expression right here is a transformation of a. So this is equal to c squared times the transformation of a. Let me do it in the same color. So what I've just showed you is, if I take the transformation of a vector being multiplied by a scalar quantity first, that that's equal to -- for this T, for this transformation that I've defined right here -- c squared times the transformation of a. And clearly this statement right here, or this choice of transformation, conflicts with this requirement for a linear transformation. If I have a c here I should see a c here. But in our case, I have a c here and I have a c squared here. So clearly this negates that statement. So this is not a linear transformation. And just to get a gut feel if you're just looking at something, whether it's going to be a linear transformation or not, if the transformation just involves linear combinations of the different components of the inputs, you're probably dealing with a linear transformation. If you start seeing things where the components start getting multiplied by each other or you start seeing squares or exponents, you're probably not dealing with a linear transformation. And then there's some functions that might be in a bit of a grey area, but it tends to be just linear combinations are going to lead to a linear transformation. But hopefully that gives you a good sense of things. And this leads up to what I think is one of the neatest outcomes, in the next video.

Contents

Definition and first consequences

Let and be vector spaces over the same field A function is said to be a linear map if for any two vectors and any scalar the following two conditions are satisfied:

additivity / operation of addition
homogeneity of degree 1 / operation of scalar multiplication

Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before or after the operations of addition and scalar multiplication.

This is equivalent to requiring the same for any linear combination of vectors, i.e. that for any vectors and scalars the following equality holds:[4][5]

Denoting the zero elements of the vector spaces and by and respectively, it follows that Let and in the equation for homogeneity of degree 1:

Occasionally, and can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of "linear". If and are considered as spaces over the field as above, we talk about -linear maps. For example, the conjugation of complex numbers is an -linear map , but it is not -linear.

A linear map with viewed as a vector space over itself is called a linear functional.[6]

These statements generalize to any left-module over a ring without modification, and to any right-module upon reversing of the scalar multiplication.

Examples

  • The zero map x ↦ 0 between two left-modules (or two right-modules) over the same ring is always linear.
  • The identity map on any module is a linear operator.
  • Any homothecy centered in the origin of a vector space, where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear.
  • For real numbers, the map xx2 is not linear.
  • For real numbers, the map xx + 1 is not linear (but is an affine transformation; y = x + 1 is a linear equation, as the term is used in analytic geometry.)
  • If A is a real m × n matrix, then A defines a linear map from Rn to Rm by sending the column vector xRn to the column vector AxRm. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the following section.
  • Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a linear operator on the space of all smooth functions.
  • The (definite) integral over some interval I is a linear map from the space of all real-valued integrable functions on I to R.
  • The (indefinite) integral (or antiderivative) with a fixed starting point defines a linear map from the space of all real-valued integrable functions on R to the space of all real-valued, differentiable, functions on R. Without a fixed starting point, an exercise in group theory will show that the antiderivative maps to the quotient space of the differentiables over the equivalence relation, "differ by a constant", which yields an identity class of the constant valued functions .
  • If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f : VW to dimF(W) × dimF(V) matrices in the way described in the sequel are themselves linear maps (indeed linear isomorphisms).
  • The expected value of a random variable (which is in fact a function, and as such a member of a vector space) is linear, as for random variables X and Y we have E[X + Y] = E[X] + E[Y] and E[aX] = aE[X], but the variance of a random variable is not linear.

Matrices

If V and W are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from V to W can be represented by a matrix.[7] This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if A is a real m × n matrix, then f(x) = Ax describes a linear map RnRm (see Euclidean space).

Let {v1, …, vn} be a basis for V. Then every vector v in V is uniquely determined by the coefficients c1, …, cn in the field R:

If f : VW is a linear map,

which implies that the function f is entirely determined by the vectors f(v1), …, f(vn). Now let {w1, …, wm} be a basis for W. Then we can represent each vector f(vj) as

Thus, the function f is entirely determined by the values of aij. If we put these values into an m × n matrix M, then we can conveniently use it to compute the vector output of f for any vector in V. To get M, every column j of M is a vector

corresponding to f(vj) as defined above. To define it more clearly, for some column j that corresponds to the mapping f(vj),

where M is the matrix of f. In other words, every column j = 1, …, n has a corresponding vector f(vj) whose coordinates a1j, …, amj are the elements of column j. A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen.

The matrices of a linear transformation can be represented visually:

  1. Matrix for relative to :
  2. Matrix for relative to :
  3. Transition matrix from to :
  4. Transition matrix from to :

The relationship between matrices in a linear transformation

Such that starting in the bottom left corner and looking for the bottom right corner , one would left-multiply—that is, . The equivalent method would be the "longer" method going clockwise from the same point such that is left-multiplied with , or .

Examples of linear transformation matrices

In two-dimensional space R2 linear maps are described by 2 × 2 real matrices. These are some examples:

  • rotation
    • by 90 degrees counterclockwise:
    • by an angle θ counterclockwise:
  • reflection
    • about the x axis:
    • about the y axis:
  • scaling by 2 in all directions:
  • horizontal shear mapping:
  • squeeze mapping:
  • projection onto the y axis:

Forming new linear maps from given ones

The composition of linear maps is linear: if f : VW and g : WZ are linear, then so is their composition gf : VZ. It follows from this that the class of all vector spaces over a given field K, together with K-linear maps as morphisms, forms a category.

The inverse of a linear map, when defined, is again a linear map.

If f1 : VW and f2 : VW are linear, then so is their pointwise sum f1 + f2 (which is defined by (f1 + f2)(x) = f1(x) + f2(x)).

If f : VW is linear and a is an element of the ground field K, then the map af, defined by (af)(x) = a(f(x)), is also linear.

Thus the set L(V, W) of linear maps from V to W itself forms a vector space over K, sometimes denoted Hom(V, W). Furthermore, in the case that V = W, this vector space (denoted End(V)) is an associative algebra under composition of maps, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.

Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication, the addition of linear maps corresponds to the matrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

Endomorphisms and automorphisms

A linear transformation f: VV is an endomorphism of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is the identity map id: VV.

An endomorphism of V that is also an isomorphism is called an automorphism of V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a group, the automorphism group of V which is denoted by Aut(V) or GL(V). Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V).

If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n × n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n × n invertible matrices with entries in K.

Kernel, image and the rank–nullity theorem

If f : VW is linear, we define the kernel and the image or range of f by

ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula is known as the rank–nullity theorem:

[8]

The number dim(im(f)) is also called the rank of f and written as rank(f), or sometimes, ρ(f); the number dim(ker(f)) is called the nullity of f and written as null(f) or ν(f). If V and W are finite-dimensional, bases have been chosen and f is represented by the matrix A, then the rank and nullity of f are equal to the rank and nullity of the matrix A, respectively.

Cokernel

A subtler invariant of a linear transformation is the cokernel, which is defined as

This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the co-kernel is a quotient space of the target. Formally, one has the exact sequence

These can be interpreted thus: given a linear equation f(v) = w to solve,

  • the kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the number of degrees of freedom in a solution, if it exists;
  • the co-kernel is the space of constraints that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution.

The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space W/f(V) is the dimension of the target space minus the dimension of the image.

As a simple example, consider the map f: R2R2, given by f(x, y) = (0, y). Then for an equation f(x, y) = (a, b) to have a solution, we must have a = 0 (one constraint), and in that case the solution space is (x, b) or equivalently stated, (0, b) + (x, 0), (one degree of freedom). The kernel may be expressed as the subspace (x, 0) < V: the value of x is the freedom in a solution – while the cokernel may be expressed via the map WR, given a vector (a, b), the value of a is the obstruction to there being a solution.

An example illustrating the infinite-dimensional case is afforded by the map f: RR, with b1 = 0 and bn + 1 = an for n > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel ( ), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an endomorphism have the same dimension (0 ≠ 1). The reverse situation obtains for the map h: RR, with cn = an + 1. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.

Index

For a linear operator with finite-dimensional kernel and co-kernel, one may define index as:

namely the degrees of freedom minus the number of constraints.

For a transformation between finite-dimensional vector spaces, this is just the difference dim(V) − dim(W), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom.

The index of an operator is precisely the Euler characteristic of the 2-term complex 0 → VW → 0. In operator theory, the index of Fredholm operators is an object of study, with a major result being the Atiyah–Singer index theorem.[9]

Algebraic classifications of linear transformations

No classification of linear maps could hope to be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.

Let V and W denote vector spaces over a field, F. Let T: VW be a linear map.

  • T is said to be injective or a monomorphism if any of the following equivalent conditions are true:
    • T is one-to-one as a map of sets.
    • kerT = {0V}
    • dim(kerT) = 0
    • T is monic or left-cancellable, which is to say, for any vector space U and any pair of linear maps R: UV and S: UV, the equation TR = TS implies R = S.
    • T is left-invertible, which is to say there exists a linear map S: WV such that ST is the identity map on V.
  • T is said to be surjective or an epimorphism if any of the following equivalent conditions are true:
    • T is onto as a map of sets.
    • coker T = {0W}
    • T is epic or right-cancellable, which is to say, for any vector space U and any pair of linear maps R: WU and S: WU, the equation RT = ST implies R = S.
    • T is right-invertible, which is to say there exists a linear map S: WV such that TS is the identity map on W.
  • T is said to be an isomorphism if it is both left- and right-invertible. This is equivalent to T being both one-to-one and onto (a bijection of sets) or also to T being both epic and monic, and so being a bimorphism.
  • If T: VV is an endomorphism, then:
    • If, for some positive integer n, the n-th iterate of T, Tn, is identically zero, then T is said to be nilpotent.
    • If T2 = T, then T is said to be idempotent
    • If T = kI, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map; see scalar matrix.

Change of basis

Given a linear map which is an endomorphism whose matrix is A, in the basis B of the space it transforms vector coordinates [u] as [v] = A[u]. As vectors change with the inverse of B (vectors are contravariant) its inverse transformation is [v] = B[v'].

Substituting this in the first expression

hence

Therefore, the matrix in the new basis is A′ = B−1AB, being B the matrix of the given basis.

Therefore, linear maps are said to be 1-co- 1-contra-variant objects, or type (1, 1) tensors.

Continuity

A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional.[10] An infinite-dimensional domain may have discontinuous linear operators.

An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example, sin(nx)/n converges to 0, but its derivative cos(nx) does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).

Applications

A specific application of linear maps is for geometric transformations, such as those performed in computer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames.

Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.

See also

Notes

  1. ^ Linear transformations of V into V are often called linear operators on V Rudin 1976, p. 207
  2. ^ Let V and W be two real vector spaces. A mapping a from V into W Is called a 'linear mapping' or 'linear transformation' or 'linear operator' [...] from V into W, if
    for all ,
    for all and all real λ. Bronshtein, Semendyayev 2004, p. 316
  3. ^ Rudin 1991, p. 14
    Here are some properties of linear mappings whose proofs are so easy that we omit them; it is assumed that and :
    1. If A is a subspace (or a convex set, or a balanced set) the same is true of
    2. If B is a subspace (or a convex set, or a balanced set) the same is true of
    3. In particular, the set:
      is a subspace of X, called the null space of .
  4. ^ Rudin 1991, p. 14. Suppose now that X and Y are vector spaces over the same scalar field. A mapping is said to be linear if for all and all scalars and . Note that one often writes , rather than , when is linear.
  5. ^ Rudin 1976, p. 206. A mapping A of a vector space X into a vector space Y is said to be a linear transformation if: for all and all scalars c. Note that one often writes instead of if A is linear.
  6. ^ Rudin 1991, p. 14. Linear mappings of X onto its scalar field are called linear functionals.
  7. ^ Rudin 1976, p. 210 Suppose and are bases of vector spaces X and Y, respectively. Then every determines a set of numbers such that
    It is convenient to represent these numbers in a rectangular array of m rows and n columns, called an m by n matrix:
    Observe that the coordinates of the vector (with respect to the basis ) appear in the jth column of . The vectors are therefore sometimes called the column vectors of . With this terminology, the range of A is spanned by the column vectors of .
  8. ^ Horn & Johnson 2013, 0.2.3 Vector spaces associated with a matrix or linear transformation, p. 6
  9. ^ Nistor, Victor (2001) [1994], "Index theory", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4: "The main question in index theory is to provide index formulas for classes of Fredholm operators ... Index theory has become a subject on its own only after M. F. Atiyah and I. Singer published their index theorems"
  10. ^ Rudin 1991, p. 15 1.18 Theorem Let be a linear functional on a topological vector space X. Assume for some . Then each of the following four properties implies the other three:
    1. is continuous
    2. The null space is closed.
    3. is not dense in X.
    4. is bounded in some neighbourhood V of 0.

References

This page was last edited on 24 June 2019, at 03:25
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