Linear algebra

I need the solution for these

Q.1 (20) Concepts and vocabulary. Circle “True” or “False”. If you choose “False”, give
reasons or a counter example to support your answer.
1. If A and B are n x n matrices, then (A+B)(A – B) = A2 – B2. (True or False)
2. If A is a non-invertible n x n matrix, then det(A) = det(rref(A)). (True or False)
3. If A and B both have as an eigenvalue, then ) is an eigenvalue of AB. (True or False)
4. U = {(x,y) € R2 : 22 + y2 S1} is a subspace of R?. (True or False)
5. If a a square matrix A has non-trivial null space, then 0 is an eigenvalue of A. (True or False)
6. If the rank of an nxn matrix A is less than n, then 0 is an eigenvalue of A. (True or False)
7. The rank of an lower-triangular matrix cquals the number of non-zero entries along the diag-
onal. (True or False)
8. For 2 x 2 matrices A and B, if AB = 0, then either A = 0 or B = 0. (True or False)
9. The rank of an m x n matrix is at most m. (True or False)
10. If A is a 4 x 3 matrix and rref(A) has exactly two nonzero rows, then dim (null(A)) = 1.
(True or False)
Q.2 (15) Determine whether (3,3, 2) is in the range of the linear transformation T: R + R3
defined by
T(a1, 22, 23) = (a1 + a2 + a3, 21 – 22 + 23, Q1 +03)
***
Dan
Q.3 (15) Let T be the linear operator on P2(R) defined by T(f(x)) = f'(x). The matrix represen-
tation of T with respect to standard basis for P2(R) is
0 1 0
0 0 2
0
0 0 0
What are the eigenvalues of T? Find the corresponding eigenspaces.
数
SN
Q.4 (10) Consider W
and dim IV-
span({ei. e2}) in F (e, and ey are members of standard basis). Find W-
66
Q.5 (10) Represent the polynomial f(x) = 1 + 2x + 3×2 as a linear combination of the vectors in
the ONB for P2(R).
Q.1 (20) Concepts and vocabulary. Circle “True” or “False”. If you choose “False”, give
reasons or a counter example to support your answer.
1. If A and B are n x n matrices, then (A+B)(A – B) = A2 – B2. (True or False)
2. If A is a non-invertible n x n matrix, then det(A) = det(rref(A)). (True or False)
3. If A and B both have as an eigenvalue, then ) is an eigenvalue of AB. (True or False)
4. U = {(x,y) € R2 : 22 + y2 S1} is a subspace of R?. (True or False)
5. If a a square matrix A has non-trivial null space, then 0 is an eigenvalue of A. (True or False)
6. If the rank of an nxn matrix A is less than n, then 0 is an eigenvalue of A. (True or False)
7. The rank of an lower-triangular matrix cquals the number of non-zero entries along the diag-
onal. (True or False)
8. For 2 x 2 matrices A and B, if AB = 0, then either A = 0 or B = 0. (True or False)
9. The rank of an m x n matrix is at most m. (True or False)
10. If A is a 4 x 3 matrix and rref(A) has exactly two nonzero rows, then dim (null(A)) = 1.
(True or False)
Q.2 (15) Determine whether (3,3, 2) is in the range of the linear transformation T: R + R3
defined by
T(a1, 22, 23) = (a1 + a2 + a3, 21 – 22 + 23, Q1 +03)
***
Dan
Q.3 (15) Let T be the linear operator on P2(R) defined by T(f(x)) = f'(x). The matrix represen-
tation of T with respect to standard basis for P2(R) is
0 1 0
0 0 2
0
0 0 0
What are the eigenvalues of T? Find the corresponding eigenspaces.
数
SN
Q.4 (10) Consider W
and dim IV-
span({ei. e2}) in F (e, and ey are members of standard basis). Find W-