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1 Winter 206

2 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u and v, and all scalars k This definition is equivalent to saing T (a u + b v) at (u) + bt (v) for all vectors u and v, and all scalars, a and b We sa, T preserves linear combinations Basic Terminolog For the linear transformation T : R m R n, R m is called the domain of T R n is called the codomain of T For the vector u R m, the vector T (u) is called the image of u under T The set of all images of vectors u R m is called the range of T, and denoted b range(t ) It is a subset of R n A linear transformations can also be called a linear map, or a linear function, or simpl a map

3 Example Define T : R m R m b T (v) 2 v Then T (u + v) 2 (u + v) 2 u + 2 v T (u) + T (v), and T (k v) 2(k v) k(2 v) kt (v) So T is a linear transformation Indeed, an scalar map, T (v) a v, for an scalar a, is a linear transformation, as ou can check Example 2 Let T : R 2 R 2 be dfined b ([ ) [ x 2x + 3 T x + [ x 2 + [ 3 Then T is a linear transformation The eas wa to see this is to rewrite T in terms of products: T (v) A v if [ [ 2 3 x A, v Then and T (u + v) A(u + v) A u + A v T (u) + T (v), T (k v) A(k v) ka v kt (v) So T is a linear transformation, known as a matrix transformation

4 Dimensions of a Matrix Here is some terminolog about matrices that ou should be aware of: Suppose A is a matrix with n rows and m columns Then we sa that A is an n m matrix, and that A has dimensions n m The dimensions of a matrix are often called its size If m n then A is called a square matrix For example, A is a 2 4 matrix and B is a 3 3 square matrix: [ A, B Matrix Transformations Theorem 32: Let A be an n m matrix; let T (x) A x Then T : R m R n is a linear transformation Proof: as in Example 2 Less obviousl, ever linear transformation is actuall a matrix transformation for some matrix A To see this, define the vectors e 0 0, e 2 in R m and consider the images 0 0,, e m T (e ), T (e 2 ),, T (e m ) 0 0

5 The Matrix of a Linear Transformation Theorem 38: Let T : R m R n be a linear transformation, let A [ T (e ) T (e 2 ) T (e m ), which is an n m matrix Then T (x) A x Proof: first observe that if x R m with components x,, x m then x x e + x 2 e x m e m, and so T (x) T (x e + x 2 e x m e m ) x T (e ) + x 2 T (e 2 ) + + x m T (e m ) A x Note: A is called the standard matrix of T, or just the matrix of T Example 3 If then T (e ) T [ 4 x z [, T (e 2 ) [ x + 2z 4x + 3z, [ 2, T (e 3 ) 3 So the matrix of T is A [ 2 4 3

6 One-to-one and Onto Linear Transformations Let T : R m R n be a linear transformation Definition : T is one-to-one if for ever vector w R n there exists at most one vector u R m such that T (u) w Definition 2: T is onto if for ever vector w R n there exists at least one vector u R m such that T (u) w ie range(t ) R n Alternate Version of Definition : T is one-to-one if T (u) T (v) u v Theorem 35: Let T be a linear transformation T is one-to-one if and onl if T (x) 0 implies that x 0 Proof: ( ) T (x) 0 and T (0) 0 impl x 0 ( ) T (u) T (v) T (u v) 0 x u v 0 u v Example 4 Let T : R 2 R 3 be defined b T ([ x ) 2x + x 2 4x + 3 T is one-to-one: T (x) 0 2x + 0, x 2 0, 4x x 0 T is not onto: T ([ x ) 2x + x 2 4x + 3 x so range(t ) is spanned b two vectors, which cannot span R 3 ;

7 Theorems About a Transformation and its Matrix Let A [ a a 2 a m be an n m matrix and suppose T : R m R n is the linear transformation defined b T (x) A x Theorem 33: (a) w range(t ) if and onl if A x w is a consistent sstem (b) range(t ) span{a, a 2,, a m } Theorem 36: (a) T is one-to-one if and onl if {a, a 2,, a m } is independent (b) If n < m, then T is not one-to-one Theorem 37: (a) T is onto if and onl if R n span{a, a 2,, a m } (b) If n > m, then T is not onto Proofs: in each case, (b) is an immediate consequence of (a) And Theorem 33 (b) immediatel implies Theorem 37 (a) So all we reall have to proof is part (a) of both Theorems 33 and (a): w range(t ) if and onl if there is vector x R m such that T (x) w A x w if and onl if the sstem A x w is a consistent sstem 36 (a): use Theorem 35 T is one-to-one if and onl if the equation T (x) 0 has onl the trivial solution if and onl if the equation A x 0 has onl the trivial solution if and onl if the columns of A are independent, b Theorem 23

8 The Big Theorem, Version 2 We can now add two more conditions to the Big Theorem: Theorem 39: Let A {a, a 2,, a n } be a set of n vectors in R n, let A [ a a 2 a n, and let T : R n R n be a linear transformation such that T (x) A x Then the following statements are equivalent: (a) A spans R n (b) A is linearl independent (c) A x b has a unique solution for all b in R n (d) T is onto (e) T is one-to-one Example 5 As in Exercise 52, page 93 of Holt, the cross product of two vectors u, v in R 3 is defined to be u v u 2 v 3 v 2 u 3 u 3 v v 3 u u v 2 v u 2, and for an vector u R 3 the map defined b T (x) u x is linear Show that T is neither one-to-one nor onto Solution: if u 0, thent (x) 0 x 0 and T is as far from onto and one-to-one as it is possible to be If u 0, then T (u) 0, impling T is not one-to-one So b the Big Theorem it isn t onto either Aside: check that the matrix of T is A [ T (e ) T (e 2 ) T (e 3 ) 0 u 3 u 2 u 3 0 u u 2 u 0

9 Geometric Interpretation of a Matrix Transformation One wa to see the geometric effect of a linear transformation T : R 2 R 2 is to look at the image of the unit square The unit square is the square determined b e and e 2 The image of the unit square is the parallelogram determined b T (e ) and T (e 2 ) T (e 2 ) e 2 T (e ) O x e Example 6: Expansion in the Direction Let T ([ x ) [ [ x [ x 3 The image of the unit square is a rectangle with area 3 T is called an expansion of factor 3 in the direction If the factor is less than one, then T would be called a compression e 2 O T (e 2 ) 3e 2 e T (e ) x

10 Example 7: Shear in the x Direction Let T ([ x ) [ 3 0 [ x [ x + 3 The image of the unit square is a parallelogram that still has area T is called a shear of factor 3 in the x direction e 2 O x e Example 8: Dilation Let T ([ x ) [ [ x [ 3x 3 The image of the unit square is a bigger square, with area 9 T is called a dilation of factor 3 If the factor is less than one, then T would be called a contraction e 2 O T (e 2 ) 3e 2 e x T (e ) 3e

11 Rotations Let T (v) be the result of rotating the vector v counterclockwise through an angle θ about the origin [ cos θ T (e ) sin θ e 2 T (e ) [ ( ) cos θ + π T (e 2 ) T (e 2 ) 2 sin ( θ + π ) θ 2 [ O e x sin θ cos θ [ cos θ sin θ So the matrix of T is R θ sin θ cos θ Example 9 The standard matrix of a rotation of π about the origin is [ [ [ cos π sin π 0 x R π and R sin π cos π 0 π [ x The standard matrix of a rotation of π 6 about the origin is R π/6 [ ( ) ( ) cos π 6 sin π 6 sin ( π ) 6 cos ( π ) 6 2 [ 3 3 and R π/6 [ x 2 [ 3 x + x + 3

12 The Standard Matrix of a Reflection in the Line mx Let d [ m Check that: T (e ) +m 2 [ m 2 2m [ T (e 2 ) +m 2 2m m 2 So the matrix of T is Q m [ m 2 2m + m 2 2m m 2 T (e ) mx d O e x T (e 2 ) 7 e 2 Example 0 The standard matrix for a reflection in the x-axis, with m 0, is [ [ [ 0 x x Q 0 and Q 0 0 The standard matrix for a reflection in the line x is Q [ [ [ [ x and Q x The standard matrix for a reflection in the line 3x is Q 3 [ 8 6 [ 4 3 and Q 3 [ x 5 [ 4x 3 3x + 4

13 Subscript Notation Up until now a matrix has been used to represent a sstem of linear equations, or to represent a linear transformation In this section we shall treat matrices as algebraic objects in their own right In general, an n m matrix A a a 2 a 3 a m a 2 a 22 a 23 a 2m a 3 a 32 a 33 a 3m a n a n2 a n3 a nm can be represented as A [a ij, where i n, and j m First definition: two matrices A and B are equal if A and B have the same dimensions and corresponding entries are equal, a ij b ij Matrix Addition If A [a ij and B [b ij are both n m matrices, then A + B and A B are defined as follows: A + B [a ij + b ij and A B [a ij b ij That is, ou add and subtract componentwise For example, if [ [ A and B then A + B [ and A B [

14 Zero Matrices An matrix for which ever entr is 0, is called a zero matrix Eg, [ is a zero matrix To be more precise, it is a 2 3 zero matrix A zero matrix is alwas represented b the number 0 with appropriate subscripts added to emphasize the size of the matrix: 0 23 [ and are both zero matrices Often the subscripts are dropped, if it is clear from context that 0 represents a matrix The Negative of a Matrix If A [a ij is an n m matrix, then A is called the negative of A; it is defined as A [ a ij For example, if then A A [ , [ Matrix subtraction is thus addition of the negative: A B A + ( B)

15 Properties of Matrix Addition Suppose A, B and C are all n m matrices Then A + B B + A A + (B + C) (A + B) + C 0 nm + A A, A + ( A) 0 nm The proofs are all eas computations; see the textbook Scalar Multiplication If A [a ij is an n m matrix, and k is a scalar, that is, a real number, then ka is called a scalar multiple of A and it is defined b ka [ka ij For example, if then A 3A [ [ ,

16 Properties of Scalar Multiplication Let A, B be n m matrices; let c and d be scalars Then c(a + B) ca + cb (c + d)a ca + da c(da) (cd)a A A 0A 0 nm, ( )A A If ca 0 nm, then c 0 or A 0 nm Again, the proofs are all eas Vectors as Matrices An n matrix is called a column matrix, or a column vector Thus vectors as defined in Chapter are just special cases of matrices However, we will still reserve the notation x x x 2 x n for vectors, although ou could just as well use x X x 2 x n

17 Transposition Let A [a ij be an n m matrix The transpose of A, denoted b A T, is the m n matrix defined b A T [a ji That is, A T is obtained from A b interchanging rows and columns For example, if [ A, 0 6 then A T Properties of Transposition Let A and B be n m matrices; let k be a scalar Then (A T ) T A (ka) T ka T (A + B) T A T + B T See the textbook for the (eas) proofs

18 Smmetric Matrices A square matrix A for which A T A is called a smmetric matrix For example: 3 6 A is smmetric, but B [ is not Example ( [ [ ) T [ T

19 Composition of Transformations and Matrix Multiplication Suppose both S : R m R k and T : R k R n are linear transformations Then the composition, T S, defined b (T S)(v) T (S(v)), is also a linear transformation from R m to R n : (T S)(au + bv) T (S(au + bv)), b definition T (as(u) + bs(v)), because S is linear a T (S(u)) + b T (S(v)), because T is linear a (T S)(u) + b (T S)(v) What is the matrix of T S in terms of the matrices of T and S? The Matrix of T S Suppose the matrix of T is the n k matrix A and the matrix of S is the k m matrix B [ b b 2 b m Then (T S)(x) T (S(x)) A(B x) A(x b + x 2 b x m b m ) x A b + x 2 A b x m a b m [ A b A b 2 A b m x The n m matrix C [ A b A b 2 A b m is the matrix of T S It is defined as the product AB : if A is n k, B is k m, then AB [ A b A b 2 A b m, which is an n m matrix

20 Example 2 Let A [ [ 2 5, B 9 4 [ 3, C Both AC and BC are defined: [ [ [ AC A A 2 4 [ BC B [ 2 B [ 4 But CA and CB are not defined [ A [ B [ [ ; Both products AB and BA are possible Let s calculate them: [ [ AB [ BA [ [ [ Note that AB BA Be careful! matrix multiplication has properties different than number multiplication

21 An Alternate Definition for the Product AB If A is an n k matrix and B is a k m matrix, then the n m matrix C [c ij, defined b c ij a i b j + a i2 b 2j + + a ik b kj, for i n, j m, is called the product of A and B, and we write C AB If ou have seen the dot product of vectors in high school, then c ij r i c j, where r i [ a i a i2 a ik and cj b j b 2j b kj are the i-th row of A and the j-th column of B, respectivel Example 3 Take A and C as in Example 2 Here is how we can calculate the product AC: [ [ AC [ 3() + 7(2) 3( ) + 7(4) 3(3) + 7( 2) 2() + 5(2) 2( ) + 5(4) 2(3) + 5( 2) [ Regardless of which wa ou calculate the product of two matrices, it is not necessar to write down ever miniscule computation: we usuall do the arithmetic in our heads

22 Identit Matrices A square matrix with ever entr on the main diagonal equal to, and ever other entr equal to 0, is called an identit matrix, and is usuall denoted b I For example, I and I [ 0 0 are both identit matrices If it is important to indicate the sizes, ou can use subscripts: I 44 and I 22, or more sloppil, I 4 and I 2 Properties of Matrix Multiplication Let c be a scalar and let A, B, C be matrices of appropriate sizes so that the following products can be formed Then: IA A and BI B, where I is an identit matrix A(BC) (AB)C A(B ± C) AB ± AC (B ± C)A BA ± CA c(ab) (ca)b A(cB) (AB) T B T A T, note the change of order! See the textbook for the proofs Note that AB BA is not one of the properties listed above

23 Example 4 Consider the matrices [ A, B 2 2 [ [ 2, C 2 Then AB [ 2 2 [ but neither A 0 nor B 0 Similarl [ [ 2 AC [ [ Thus it is possible that AB AC but neither B C nor A 0 Example 5 Suppose T is a reflection in the line x, and S is a reflection in the line 2x Then the matrix of T is Q [ [ [ and the matrix of S is Q The matrix of T S is 4 3 Q Q 2 [ [ [ 4 3 R cos ( 4/5) and (T S)(v) 5 [ [ x 5 [ 4x 3 3x 4

24 Powers of a Matrix If A is a square matrix we define A 2 AA, A 3 AAA, etc [ 3 7 A 2 5 So for ou can check that A 2 [ [ [ and A 3 A 2 B [ [ [ It can be ver cumbersome to calculate higher powers of a matrix! Example 6 For numbers, x 2 x ± Things are ver different for matrices Consider the matrix equation A 2 I Two obvious solutions are A ±I, but there are infinitel man more For example, if A is 2 2: [ 0 /c A, for c 0 c 0 Can ou find an others?

25 Diagonal and Triangular Matrices D U L a a a nn a a 2 a n 0 a 22 a 2n 0 0 a nn a 0 0 a 2 a 22 0 a n a n2 a nn D [a ij is diagonal if a ij 0, i j U [a ij is upper triangular if a ij 0, i > j L [a ij is lower triangular if a ij 0, i < j Powers of Diagonal and Triangular Matrices Powers of these tpes of matrices are of the same form That is D k is diagonal if D is; U k is upper triangular if U is; and L k is lower triangular if L is For example: D D k 4 k k ( 3) k and U U , U , as ou can check

26 Block Multiplication Sometimes it is convenient to partition the entries of a matrix into blocks Consider the matrix [ A P Q, O R where the 2 2 matrix P, the 2 3 matrix Q, the 2 matrix O, and the 3 matrix R are all called blocks If B [ U V, where U and V are also blocks, then AB can be calculated as [ [ P Q U AB O R V [ PU + QV OU + RV [ PU + QV RV , as ou ma check 49 5

27 Inverse Linear Transformations The linear transformation T : R n R n is called invertible if it is one-to-one and onto If T is invertible, the inverse function T : R n R n is defined b T () x T (x) That is (T T )() and (T T )(x) x, for all x, R n Note: The function T is well defined because For each R n there is at least one x R n such that T (x), because T is onto 2 For each R n there is at most one x R n such that T (x), because T is one-to-one Thus for each R n there is exactl one x R n such that T (x), and so T is well defined T is Also a Linear Transformation Theorem 39: If T : R n R n is invertible, then T is also a linear transformation Proof: let, 2 be vectors in R n, let a, a 2 be scalars Since T is invertible there are unique vectors x, x 2 R n such that Then T (x ) and T (x 2 ) 2 T (a + a 2 2 ) T (a T (x ) + a 2 T (x 2 )) T (T (a x + a 2 x 2 )) (wh?) a x + a 2 x 2 a T ( 2 ) + a 2 T ( 2 )

28 The Matrix of T Suppose that T : R n R n is invertible, the matrix of T is A, and the matrix of T is B Then for all R n, This means that T (T ()) T (B ) AB AB I Definition: the n n matrix A is invertible if there is an n n matrix B such that AB I Example Suppose that T : R 2 R 2 is a shear of factor 2 in the x-direction What is its inverse? [ [ 2 2 Soluton: The matrix of T is A ; let B 0 0 Then [ [ [ AB So the matrix of T is B, and [ [ T 2 x (v) 0 [ x 2 That is, T is a shear of factor 2 in the negative x-direction

29 The Inverse of a Matrix Theorem 32: Suppose A is an n n invertible matrix and B is an n n matrix such that AB I Then BA I, and if C is another n n matrix such that AC I, then C B Proof: Let e i, a i, b i, for i n, be the i-th column of the matrices I, A, B, respectivel I AB e i AB e i A b i, so R n span{e,, e n } span{a b,, A b n } span{a,, a n } Thus the columns of A span R n, and b the Big Theorem, the onl solution to A 0 is the trivial solution Now for an x R n : I (Ax) Ax (AB)(Ax) Ax A(BA)x Ax A(BAx x) 0 Thus BAx x, for all x, and it follows that BA I Now suppose AB I and AC I Then B(AB) B(AC) (BA)B (BA)C IB IC B C So for an invertible matrix A there is a unique matrix B such that AB I BA This matrix B is called the inverse of A and thus: Definitions: If A is an n n invertible matrix, A is called the inverse of A and denotes the unique matrix such that AA I A A (In other books ou ma see a one-step definition: A is invertible if there is a matrix B, called A, such that AB I BA) Also: another word for invertible is nonsingular Then a matrix is singular if it is not invertible

30 Properties of the Inverse Theorem 323 Extended: Let A, B, A,, A k be n n invertible matrices, let C, D be n m matrices Then: A is invertible and ( A ) A 2 A p is invertible for p, and (A p ) ( A ) p 3 A T is invertible and ( A T ) ( A ) T 4 If c 0, then ca is invertible and (ca) c A 5 AB is invertible and (AB) B A 6 A A 2 A k is invertible and (A A 2 A k ) A k 7 If AC AD then C D 8 If AC 0 nm then C 0 nm A 2 A Proof: here are some You can fill in the rest 3 AA I (AA ) T I T (A ) T A T I (A T ) (A ) T 5: check the given formula: AB(B A ) A(BB )A AIA AA I 7: AC AD A AC A AD IC ID C D 8 AC 0 nm A AC A 0 nm IC 0 nm C 0 nm The question remains: how to compute A? There turn out to be various was, but the most basic is called the Gaussian algorithm

31 The Gaussian Algorithm for Finding Matrix Inverses If AB I, then ABe i e i A b i e i So solving the sstem A b i e i will give ou b i, which is the i-th column of B A But instead of solving n separate sstems, all with the same coefficient matrix A, use row reduction on the augmented matrix (A I ) : reduce the left side until it becomes I, if possible Appl the same operations to the right side; when the left side becomes I the right side will become A That is (A I ) ( I A ) This method is actuall the most efficient wa for finding inverses If it is not possible to reduce A to I on the left side, then A is not invertible Example 2 A [A I (I A ) A

32 Example 3 Let then [A I A , So A [ I A

33 Example 4 Consider the sstem of equations in three variables x, x 2, x 3 : 2x + x 2 x 3 x + x 2 + x x 2 + 4x 3 3 This can be written as a single matrix equation A x b, with 2 x A, x x 2, b x 3 3 From Example 3, A It can be used to solve the sstem of equations in a different wa: A x b A (A x) A b (A A)x A b x A b That is, the solution to the sstem is x x x

34 The Inverse of a 2 2 Matrix Let [ a b A c d be the general 2 2 matrix Then [ A d b ad bc c a, if ad bc 0 Verif: ad bc [ [ d b a b ad bc c a c d [ da bc db bd ca + ac cb + ad [ 0 0 I Example 5 Let T : R 2 R 2 be a rotation of θ counterclockwise around the origin Then the matrix of T is [ cos θ sin θ R θ sin θ cos θ and the matrix of T is R [ cos θ cos 2 θ + sin 2 θ sin θ θ sin θ cos θ [ cos( θ) sin( θ) sin( θ) cos( θ) R θ That is, the inverse of a rotation of θ counterclockwise around the origin is a rotation of θ clockwise around the origin

35 Invertible Coefficient Matrices and Sstems of Equations Theorem 324: Let A be an n n matrix Then the following statements are equivalent A is invertible 2 A x b has a unique solution for all b R n : x A b 3 A x 0 has onl the trivial solution Proof: as in Example 4, it is simple calculation to show that () (2) (3) To complete the ccle of implications we need to show (3) () : if the onl solution to A x 0 is x 0, then the reduced echelon form of A must be I B the Gaussian algorithm, A exists The Big Theorem, Version 3 Theorem 325: Let A {a, a 2,, a n } be a set of n vectors in R n, let A [ a a 2 a n, and let T : R n R n be a linear transformation such that T (x) A x Then the following statements are equivalent: (a) A spans R n (b) A is linearl independent (c) A x b has a unique solution for all b in R n (d) T is onto (e) T is one-to-one (f) A is invertible

36 When are Diagonal or Triangular Matrices Invertible? One consequence of the Big Theorem is that a triangular matrix A is invertible if and onl if the product of its diagonal entries is not zero (Because its columns must be independent) That is, A is invertible a a 22 a 33 a nn 0 and it is possible b inspection to tell if a triangular matrix diagonal, upper triangular, or lower triangular is invertible So if U 0 2 9, L 3 0 0, D 0 5 0, then U and D are invertible, but L is not Inverses of Diagonal or Triangular Matrices Invertible The inverses of these matrices are of the same form That is, the inverse of an invertible diagonal matrix is another diagonal matrix, the inverse of an invertible upper triangular matrix is another upper triangular matrix, and the inverse of an invertible lower triangular matrix is another lower triangular matrix For example, with U and D as above: U , D

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