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Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Linear Transformations and Matrix Algebra A. Havens Department of Mathematics University of Massachusetts, Amherst February 10-16, 2018 A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Outline 1 Representing Linear Maps with Matrices n The Standard Basis of R Finding Matrices Representing Linear Maps 2 Existence/Uniqueness Redux Reframing via Linear Transformations Surjectivity, or Onto Maps Injectivity, or One-To-One Maps Theorems on Existence and Uniqueness 3 Matrix Algebra Composition of Maps and Matrix Multiplication Matrices as Vectors: Scaling and Addition Transposition A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra n The Standard Basis of R Components Revisited 2 Observe that any x 2 R can be written as a linear combination of vectors along the standard rectangular coordinate axes using their components relative to this standard rectangular coordinate system: ñ ô ñ ô ñ ô x1 1 0 x = = x1 + x2 : x2 0 1 These two vectors along the coordinate axes will form the standard 2 basis for R . A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra n The Standard Basis of R Elementary Vectors Definition 2 The vectors along the standard rectangular coordinate axes of R are denoted ñ 1 ô ñ 0 ô e := ; e := : 1 0 2 1 They are called elementary vectors (hence the notation ei , i = 1; 2), and the ordered list (e1; e2) is called the standard basis 2 of R . 2 Observe that Span fe1; e2g = R . A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra n The Standard Basis of R Elementary Vectors n We can also define elementary vectors and a standard basis in R , by taking the unit vectors along the n different coordinate axes of the standard rectangular coordinate system: Definition The n vectors 2 1 3 2 0 3 2 0 3 2 0 3 6 0 7 6 1 7 6 . 7 6 . 7 6 7 6 7 6 . 7 6 . 7 6 7 6 7 6 7 6 7 e1 := 6 0 7 ; e2 := 6 0 7 ;:::; en−1 := 6 7 ; en := 6 7 6 7 6 7 6 0 7 6 0 7 6 . 7 6 . 7 6 7 6 7 4 . 5 4 . 5 4 1 5 4 0 5 0 0 0 1 are called elementary vectors for the standard rectangular n coordinate system on R . A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra n The Standard Basis of R A Remark on Notation Remark Note that we use the symbols e1; e2;::: to respectively represent the first, second, etc elementary vectors in whatever real vector space we are working in, indexed with respect to the order of our coordinate axes. The number of components necessary to represent a given ei n depends on the particular R with which we are working, and will be clear from context. Thus, the notation ei always refers to a vector with a 1 in the ith component, but the vector may have however many zeroes we need. A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra n The Standard Basis of R The Standard Basis of Rn Observation n Clearly, the set fe1;:::; eng ⊂ R is linearly independent, as the matrix [e1 ::: en] has precisely n columns and n pivots. Definition The ordered n-tuple of vectors (e1;:::; en) is called the standard n basis of R . n X n Observe that x = xi ei =) Span fe1;:::; eng = R . i=1 2 Therefore in analogy to the case of R , the n-tuple (e1;:::; en) earns the title of a basis because it is an ordered, linearly n independent collection of vectors that spans the whole of R . A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra n The Standard Basis of R RREF and the Standard Basis Observation Given a collection of n linearly independent vectors v1 ;:::; vn, the matrix with these vectors as columns has 2 1 0 ::: 0 3 6 7 6 0 1 ::: 0 7 RREF[v1 ::: vn] = [e1 ::: en] = 6 . 7 : 6 . .. 7 4 . 5 0 ::: 0 1 Thus, any n × n matrix whose columns are linearly independent is row equivalent to the matrix whose columns are the standard basis. A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Finding Matrices Representing Linear Maps Matrix Representations Definition n m Given a linear map T : R ! R , we will say that an m × n matrix A is a matrix representing the linear transformation T if the image n of a vector x in R is given by the matrix vector product T (x) = Ax : Our aim is to find out how to find a matrix A representing a linear transformation T . In particular, we will see that the columns of A come directly from examining the action of T on the standard basis vectors. Before we state the formal result, let us consider a simple two dimensional reflection, and try to represent it as a matrix-vector product. A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Finding Matrices Representing Linear Maps An Example Example 2 2 Find the matrix representing the map M : R ! R that reflects a vector x through the line Span fe1 + e2g. First, note that ñ 1 ô e + e = ; 1 2 1 so Span fe1 + e2g is the solution set of the linear equation x1 − x2 = 0, i.e., it is the line x1 = x2. You should convince yourself that reflection through this line swaps the vectors e1 and e2, and in general acts on a vector by swapping its components. A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Finding Matrices Representing Linear Maps An Example Example So M(e1) = e2 and M(e2) = e1. If we consider an arbitrary 2-vector x = x1e1 + x2e2, it is easy to check that because M is linear and swaps the elementary vectors, M must swap the components of x. Indeed by linearity M(x) = M(x1e1 + x2e2) = x1M(e1) + x2M(e2) = x1e2 + x2e1. A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Finding Matrices Representing Linear Maps An Example Example Comparing the image ñ x ô M(x) = 2 x1 with ñ a b ô ñ x ô ñ ax + bx ô 1 = 1 2 c d x2 cx1 + dx2 we see that ñ x ô ñ 0x + 1x ô ñ 0 1 ô ñ x ô 2 = 1 2 = 1 : x1 1x1 + 0x2 1 0 x2 A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Finding Matrices Representing Linear Maps An Example Example So we can represent the reflection map x 7! M(x) by the matrix-vector product map ñ 0 1 ô M(x) = x = [e e ]x : 1 0 2 1 It is not a coincidence that the matrix of M is [e2 e1] = [M(e1) M(e2)]! Indeed, consider the matrix vector product Aei for an arbitrary m × n matrix A and ei the ith elementary vector of the standard n basis of R . What is the vector Aei ? A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Finding Matrices Representing Linear Maps Selecting the Matrix Columns Since ei has a one in the ith coordinate, and zeroes in all other coordinates, we deduce that Aei is the linear combination 0a1 + ::: + 0ai−1 + 1a1 + 0ai+1 + ::: + 0an = ai ; that is, Aei is just the ith column of A. If T is some linear map, and A is a matrix representing it, then we can deduce that the image of an elementary vector ei under the map T is T (ei ) = ai , so the columns of the matrix are precisely the images of the standard basis by the map T ! A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Finding Matrices Representing Linear Maps The Theorem Theorem n m A linear transformation T : R ! R may be uniquely represented as a matrix-vector product T (x) = Ax for the m × n matrix A whose columns are the images of the standard basis (e1;:::; en) of n R by the transformation T . Specifically, the ith column of A is m the vector T (ei ) 2 R and î ó T (x) = Ax = T (e1) T (e2) ::: T (en) x : A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Finding Matrices Representing Linear Maps A Satisfying, Simple Proof Proof. The result is a consequence of the calculation ÄPn ä Pn T (x) = T i=1xi ei = i=1xi T (ei ) î ó = T (e1) T (e2) ::: T (en) x =: Ax ; where the first equality follows from the representation of x in the standard basis, the second equality follows from properties of linearity, and the third equality follows from the definition of the matrix vector product Ax as being the linear combination of column vectors of A taking the components xi as the weights. A. Havens Linear Transformations and Matrix Algebra Representing Linear Maps with Matrices Existence/Uniqueness Redux Matrix Algebra Finding Matrices Representing Linear Maps Using this Result There are two ways in which this result is useful: Given a linear map described geometrically, one can examine its effect on basis elements ei and then describe the matrix representing the map, Given a matrix, one can try to understand the geometry of the map x 7! Ax by examining the columns, and understanding how the matrix acts on the frame (e1;:::; en).
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