Properties of Linear Transformations, Matrices 2

Linear Map and Values on Basis Theorem (2.6) Let (v1 , · · · , vn ) be a basis of V and (w1 , · · · , wn ) be an arbitrary list of vectors in W . Then there exists a unique linear map T : V → W such that T (vi ) = wi , ∀i = 1, 2, · · · , n. Corollary Suppose {v1 , · · · , vn } is a finite basis for V , then if U, T : V → W are linear and U(vi ) = T (vi ) for i = 1, · · · , n, then U = T . Math 4377/6308, Advanced Linear Algebra Spring, 2015 17 / 1 Matrix Transformations: Example Example      1 0 1 0 Let e1 = , e2 = , y1 =  0  and y2 =  1 . 0 1 2 1 2 3 Suppose T : R → R is a linear transformation which  maps e1   3 x1 into y1 and e2 into y2 . Find the images of and . 2 x2    Solution: First, note that T (e1 ) = and T (e2 ) = . Also  e1 + e2 = 3 2  Math 4377/6308, Advanced Linear Algebra Spring, 2015 18 / 1 Matrix Transformations: Example (cont.) Then  T 3 2  =T( T (e1 ) + e1 + e2 ) = T (e2 ) = T (3e1 + 2e2 ) = 3T (e1 ) + 2T (e2 ) Math 4377/6308, Advanced Linear Algebra Spring, 2015 19 / 1 Matrix Transformations: Example (cont.) Also  T x1 x2  =T( T (e1 ) + e1 + e2 ) = T (e2 ) = Math 4377/6308, Advanced Linear Algebra Spring, 2015 20 / 1 Coordinate Vectors Definition For a finite-dimensional vector space V , an ordered basis for V is a basis for V with a specific order. In other words, it is a finite sequence of linearly independent vectors in V that generates V . Definition Let β = {u1 , · · · , un } be an ordered basis for V , and for x ∈ V let a1 , · · · , an be the unique scalars such that x= n X ai ui . i=1 The coordinate vector of x relative to β is   a1  ..  [x]β =  .  an Math 4377/6308, Advanced Linear Algebra Spring, 2015 21 / 1 Coordinate Systems: Example Example  3 1  3 1  1 0   Let β = {b1 , b2 } where b1 = and b2 =     1 0 E = {e1 , e2 } where e1 = and e2 = . 0 1 0 1  and let Solution:  If [x]β =  If [x]E =  2 , then x = 3   6 , then x = 5   +  + 0 1  0 1    .   = = Math 4377/6308, Advanced Linear Algebra . Spring, 2015 22 / 1 Coordinate Systems: Example (cont.) Standard graph paper β− graph paper Math 4377/6308, Advanced Linear Algebra Spring, 2015 23 / 1 Matrix Representations Definition Suppose V , W are finite-dimensional vector spaces with ordered bases β = {v1 , · · · , vn }, γ = {w1 , · · · , wm }. For linear T : V → W , there are unique scalars aij ∈ F such that T (vj ) = m X aij wi for 1 ≤ j ≤ n. i=1 The m × n matrix A defined by Aij = aij is the matrix representation of T in the ordered bases β and γ, written A = [T ]γβ . If V = W and β = γ, then A = [T ]β . Note that the jth column of A is [T (vj )]γ , and if [U]γβ = [T ]γβ for linear U : V → W , then U = T . Math 4377/6308, Advanced Linear Algebra Spring, 2015 24 / 1 Matrix of Linear Transformation: Example Example      ? ?  x1 − 2x2  ? ?  x1 =   4x1 x2 ? ? 3x1 + 2x2 Solution:   ? ?  ? ?  = standard matrix of the linear transformation T ? ?   ? ?    ? ?  = T (e1 ) T (e2 ) = ? ? Math 4377/6308, Advanced Linear Algebra (fill-in) Spring, 2015 25 / 1 Matrix of Linear Transformation: Example Example Find the standard matrix of the linear transformation T : R2 → R2 which rotates a point about the origin through an angle of π4 radians (counterclockwise). ↓     T (e1 ) =   T (e2 ) =     A=  Math 4377/6308, Advanced Linear Algebra Spring, 2015 26 / 1 Identity Matrix Identity Matrix In is an n × n matrix with 1’s on the main left to right diagonal and 0’s elsewhere. The ith column of In is labeled ei . Example  I3 =  e1 e2 e3   1 0 0 = 0 1 0  0 0 1 Note that    1 0 0 x1 I3 x =  0 1 0   x2  x3 0 0 1       + =  =  +  Math 4377/6308, Advanced Linear Algebra . Spring, 2015 27 / 1 Addition and Scalar Multiplication Definition Let T , U : V → W be arbitrary functions of vector spaces V , W over F . Then T + U, aT : V → W are defined by (T + U)(x) = T (x) + U(x) and (aT )(x) = aT (x), respectively, for all x ∈ V and a ∈ F . Theorem (2.7) With the operations defined above, for vector spaces V , W over F and linear T , U : V → W : (a) aT + U is linear for all a ∈ F (b) The collection of all linear transformations from V to W is a vector space over F Definition For vector spaces V , W over F , the vector space of all linear transformations from V into W is denoted by L(V , W ), or just L(V ) if V = W . Math 4377/6308, Advanced Linear Algebra Spring, 2015 28 / 1 Matrix Representations Theorem (2.8) For finite-dimensional vector spaces V , W with ordered bases β, γ, and linear transformations T , U : V → W : (a) [T + U]γβ = [T ]γβ + [U]γβ . (b) [aT ]γβ = a[T ]γβ for all scalars a. Math 4377/6308, Advanced Linear Algebra Spring, 2015 29 / 1