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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) € RR : 2 + y2 < 1} is a subspace of R². (True or False) 5. If 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 n X n matrix A is less than n, then 0 is an eigenvalue of A. (True or False) 7. The rank of an lower-triangular matrix equals the number of non-zero entrics 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)) = (True or False) 1. Q.2 (15) Determine whether (3,3,2) is in the range of the linear transformation T : R³ R³ defined by T(a1,a2.a3) = (a1+a2+a3,a1 - a2 + a3, a1 + a3) Q.3 tation (15) of T Let with T be respect the linear to standard operator basis on P2 for (R) P2 defined (IR) is by T(f(x)) = f'(x). The matrix represen- 0 1 0 0 0 2 0 0 0 What are the eigenvalues of T? Find the corresponding eigenspaces. first two Q.4 and (10) Consider W = span ({e1,e2}) in F3 (e1 and e2 are members of standard basis). Find W- dim W. 1. m / who = Q.5 (10) Represent the polynomial f (x) = 1 + 2.c + 3x2 as a linear combination of the vectors in the ONB for P2 (R). Q.7 (10) Consider a linear transformation T : R² R³, which satisfies 1 3 1 I = 2 and T = 1 1 (a) Find the matrix A of this linear transformation. (b) Compute its reduced row-echclon form E = rref(A). (c) Does there exist a vector X E R² such that 12 T(X ) = 4 ? 3 Explain. Q.8 (10) Show that a basis for Rn cannot have more than n vectors.

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