(a) Which of the following statements are TRUE? (1) If T : R" –→ R" is a linear transformation, then N(T) is a subspace of R". (2) If Vi and V2 are subsets of F" span the same subspace S of F", then Vị = V2. (3) Let T be a linear operator on F" and let Bị be a basis for F". If Bz is another basis for F", then Ar], (t) = A(7]B, (t). Answer for (a): (b) Which of the following statements are TRUE? (1) If 0 is an eigenvalue of A E Mnxn(F), then A? = A(A) is singular. (2) Let W1 and W2 be subspaces of F", then W1 W2 is a subspace of F". (3) Let A E Mnxn(F). If AP = O for some positive integer p, then det(A) = 0.
(a) Which of the following statements are TRUE? (1) If T : R" –→ R" is a linear transformation, then N(T) is a subspace of R". (2) If Vi and V2 are subsets of F" span the same subspace S of F", then Vị = V2. (3) Let T be a linear operator on F" and let Bị be a basis for F". If Bz is another basis for F", then Ar], (t) = A(7]B, (t). Answer for (a): (b) Which of the following statements are TRUE? (1) If 0 is an eigenvalue of A E Mnxn(F), then A? = A(A) is singular. (2) Let W1 and W2 be subspaces of F", then W1 W2 is a subspace of F". (3) Let A E Mnxn(F). If AP = O for some positive integer p, then det(A) = 0.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 43EQ
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No justification is required just tell me true and false
![(a) Which of the following statements are TRUE?
(1) If T : R" → R" is a linear transformation, then N(T) is a subspace of R".
(2) If Vi and V2 are subsets of F" span the same subspace S of F", then Vị = V2.
(3) Let T be a linear operator on F" and let Bị be a basis for F".
If Bz is another basis for F", then Ar], (t) = A(7]», (t).
Answer for (a):
(b) Which of the following statements are TRUE?
(1) If 0 is an eigenvalue of A E Mnxn(F), then A? = A(A) is singular.
(2) Let W1 and W2 be subspaces of F", then W1n W2 is a subspace of F".
(3) Let A E Mnxn(F). If AP = O for some positive integer p, then det(A) = 0.
Answer for (b):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3caad12-585a-4316-b9cd-49a0c9a633d0%2F5345db74-aab8-4409-bc87-ec017d61c8cb%2Fjtcezg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Which of the following statements are TRUE?
(1) If T : R" → R" is a linear transformation, then N(T) is a subspace of R".
(2) If Vi and V2 are subsets of F" span the same subspace S of F", then Vị = V2.
(3) Let T be a linear operator on F" and let Bị be a basis for F".
If Bz is another basis for F", then Ar], (t) = A(7]», (t).
Answer for (a):
(b) Which of the following statements are TRUE?
(1) If 0 is an eigenvalue of A E Mnxn(F), then A? = A(A) is singular.
(2) Let W1 and W2 be subspaces of F", then W1n W2 is a subspace of F".
(3) Let A E Mnxn(F). If AP = O for some positive integer p, then det(A) = 0.
Answer for (b):
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