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7. Suppose T: R¹0 R¹0 is a linear map. a) If dim (N(T)) = 8, then find dim(R(T)). b) If dim(R(T)) = 10, explain why the transformation is one to one.

7. Suppose T: R¹0 R¹0 is a linear map. a) If dim (N(T)) = 8, then find dim(R(T)). b) If dim(R(T)) = 10, explain why the transformation is one to one.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.4: The Singular Value Decomposition
Problem 54EQ
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halt x 2y 1. Let T : R² → R² be given by T ([:]) = [² +2²] 3y los evitaln Prove that T is a linear operator using the definition on page 202. 2. Let T: R² →→ R2 be a linear operator. If it is known that 5 2 T ([8]) - [3] and T ([i]) - [1 = [] compute T ([ 10 2 3. Suppose T: M2x2 → M2x2 and S : M2x2 → M2x2 are linear operators where T(A) 5 1 At + A and S(A) = A - At. For the matrix A = find 6 0 a) (T+S)(A) b) (S ○ T)(A) O 4. Find a basis for the null space of the linear transformation T: R³ R³ given by (ED) T 5. Determine if the linear mapping, is an isomorphism. Explain. 6. Determine if the vector v = T ( [;)) - [² + ] x y 5y = 4 10 -3x +2y + 2z 4x + 5y + z 2y + z 5 T ( [;)) - [ H] [:], 5 1 Y 1)T qam is in R(T) for T : R² → R² given by 7. Suppose T: R¹⁰ → R¹⁰ is a linear map. a) If dim(N(T)) = 8, then find dim(R(T)). b) If dim(R(T)) = 10, explain why the transformation is one to one. =
Transcribed Image Text:halt x 2y 1. Let T : R² → R² be given by T ([:]) = [² +2²] 3y los evitaln Prove that T is a linear operator using the definition on page 202. 2. Let T: R² →→ R2 be a linear operator. If it is known that 5 2 T ([8]) - [3] and T ([i]) - [1 = [] compute T ([ 10 2 3. Suppose T: M2x2 → M2x2 and S : M2x2 → M2x2 are linear operators where T(A) 5 1 At + A and S(A) = A - At. For the matrix A = find 6 0 a) (T+S)(A) b) (S ○ T)(A) O 4. Find a basis for the null space of the linear transformation T: R³ R³ given by (ED) T 5. Determine if the linear mapping, is an isomorphism. Explain. 6. Determine if the vector v = T ( [;)) - [² + ] x y 5y = 4 10 -3x +2y + 2z 4x + 5y + z 2y + z 5 T ( [;)) - [ H] [:], 5 1 Y 1)T qam is in R(T) for T : R² → R² given by 7. Suppose T: R¹⁰ → R¹⁰ is a linear map. a) If dim(N(T)) = 8, then find dim(R(T)). b) If dim(R(T)) = 10, explain why the transformation is one to one. =
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