Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015

Topic Overview Relationship between the properties of a matrix transformation and the invertibility of the matrix Properties of linear transformations Geometric properties of eigenvectors

One-to-One Linear Transformations Definition A linear transformation T : R n R m is one-to-one if for a, b R n with a b, then T (a) T (b). [ ] cos θ sin θ Rotation operator R = is one-to-one. sin θ cos θ [ ] 1 0 Projection operator P = is not one-to-one. 0 0

One-to-One Linear Transformations Definition A linear transformation T : R n R m is one-to-one if for a, b R n with a b, then T (a) T (b). [ ] cos θ sin θ Rotation operator R = is one-to-one. sin θ cos θ [ ] 1 0 Projection operator P = is not one-to-one. 0 0

Equivalent Statements Recall: the following statements have been shown to be equivalent: A is invertible. Ax = b is consistent for every b. Ax = b has exactly one solution for every b. Think of T A as multiplication by matrix A, then the following statements are equivalent. A is invertible. For every b R n, there is some vector x R n such that T A (x) = b, i.e., the range of T A is all of R n. For every b R n, there is exactly one vector x R n such that T A (x) = b, i.e., T A is one-to-one.

Equivalent Statements Recall: the following statements have been shown to be equivalent: A is invertible. Ax = b is consistent for every b. Ax = b has exactly one solution for every b. Think of T A as multiplication by matrix A, then the following statements are equivalent. A is invertible. For every b R n, there is some vector x R n such that T A (x) = b, i.e., the range of T A is all of R n. For every b R n, there is exactly one vector x R n such that T A (x) = b, i.e., T A is one-to-one.

Summary Theorem If A is an n n matrix and T A : R n R n is multiplication by A, then the following statements are equivalent. 1. A is invertible. 2. The range of T A is R n. 3. T A is one-to-one.

s Show the rotation operator is invertible and hence it is one-to-one and onto R 2. Show the projection operator is not invertible.

s Show the rotation operator is invertible and hence it is one-to-one and onto R 2. Show the projection operator is not invertible.

Inverse of a One-to-One Operator If T A : R n R n is a one-to-one operator then matrix A is invertible. The operator T A 1 : R n R n is a linear operator called the inverse of T A. T A (T A 1(x)) = AA 1 x = I n x = x T A 1(T A (x)) = A 1 Ax = I n x = x If we use the symbol [T ] to denote the standard matrix of linear transformation T then when T is invertible, [ T 1] = [T ] 1.

Inverse of a One-to-One Operator If T A : R n R n is a one-to-one operator then matrix A is invertible. The operator T A 1 : R n R n is a linear operator called the inverse of T A. T A (T A 1(x)) = AA 1 x = I n x = x T A 1(T A (x)) = A 1 Ax = I n x = x If we use the symbol [T ] to denote the standard matrix of linear transformation T then when T is invertible, [ T 1] = [T ] 1.

Find the standard matrix for the inverse of the rotation operator.

Linearity Properties Theorem A transformation T : R n R m is linear if and only if the following relationships hold for all vectors u and v in R n and every scalar c. 1. T (u + v) = T (u) + T (v). 2. T (cu) = ct (u). Proof.

Linearity Properties Theorem A transformation T : R n R m is linear if and only if the following relationships hold for all vectors u and v in R n and every scalar c. 1. T (u + v) = T (u) + T (v). 2. T (cu) = ct (u). Proof.

Standard Basis Vectors Consider the standard basis vectors for R n. 1 0 0 e 1 =, e 1 2 =,, e n =. 0 Recall that if x R n then. 0 x = x 1 e 1 + x 2 e 2 + + x n e n, and T (x) = x 1 T (e 1 ) + x 2 T (e 2 ) + + x n T (e n ). 0 0. 1

Finding the Standard Matrix for a Transformation Theorem If T : R n R m is a linear transformation and e 1, e 2,..., e n are the standard basis vectors for R n then [T ] = [ T (e 1 ) T (e 2 ) T (e n ) ]. Find the standard matrix for the linear operator which reflects vectors about the line y = x in R 2.

Finding the Standard Matrix for a Transformation Theorem If T : R n R m is a linear transformation and e 1, e 2,..., e n are the standard basis vectors for R n then [T ] = [ T (e 1 ) T (e 2 ) T (e n ) ]. Find the standard matrix for the linear operator which reflects vectors about the line y = x in R 2.

Geometric Interpretation of Eigenvectors Recall: λ is an eigenvalue of the n n matrix A provided there exits a nontrivial solution to the linear system Ax = λx (λi A)x = 0. Definition If T : R n R n is a linear operator, then λ is an eigenvalue of T if there is a nontrivial x R n such that T (x) = λx. The nonzero vectors x that satisfy this equation are called the eigenvectors of T corresponding to λ. Remark: the eigenvalues and eigenvectors of T are precisely those of [T ].

Geometric Interpretation of Eigenvectors Recall: λ is an eigenvalue of the n n matrix A provided there exits a nontrivial solution to the linear system Ax = λx (λi A)x = 0. Definition If T : R n R n is a linear operator, then λ is an eigenvalue of T if there is a nontrivial x R n such that T (x) = λx. The nonzero vectors x that satisfy this equation are called the eigenvectors of T corresponding to λ. Remark: the eigenvalues and eigenvectors of T are precisely those of [T ].

Geometric Interpretation of Eigenvectors Recall: λ is an eigenvalue of the n n matrix A provided there exits a nontrivial solution to the linear system Ax = λx (λi A)x = 0. Definition If T : R n R n is a linear operator, then λ is an eigenvalue of T if there is a nontrivial x R n such that T (x) = λx. The nonzero vectors x that satisfy this equation are called the eigenvectors of T corresponding to λ. Remark: the eigenvalues and eigenvectors of T are precisely those of [T ].

s Find the eigenvalues and corresponding eigenvectors of the linear operator which reflects vectors about the line y = x in R 2. Find the eigenvalues and corresponding eigenvectors of the linear operator which projects vectors into the xy-plane in R 3.

s Find the eigenvalues and corresponding eigenvectors of the linear operator which reflects vectors about the line y = x in R 2. Find the eigenvalues and corresponding eigenvectors of the linear operator which projects vectors into the xy-plane in R 3.

Summary Theorem If A is an n n matrix and if T A : R n R n is multiplication by A, then the following are equivalent. 1. A is invertible. 2. Ax = 0 has only the trivial solution. 3. The reduced row echelon form of A is I n. 4. A is expressible as the product of elementary matrices. 5. Ax = b is consistent for every n 1 matrix b. 6. Ax = b has exactly one solution for every n 1 matrix b. 7. det(a) 0. 8. The range of T A is R n. 9. T A is one-to-one.

Homework Read Section 4.3 and work exercises 1 3, 5 15 odd, 21.