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0 11. Let A = -1 -3 wuNO WNWL ALNL 2 3 -2 2 a. Find Nul A. b. Find Col A. c. Find Nul AT d. Find Col AT. 12. Let A be the matrix in problem 1 1. a. Find a basis for Nul A and give the dimension of Nul A. b. Find a basis for Col A and give the dimension of Col A. Find a basis for Nul AT and give the dimension of Nul AT. d. Find a basis for Col AT and give the dimension of Col AT. 13. Suppose U = : a, b, c, d E R, 2a - 3b = c and a + b = 2c + d a. Show that U is a subspace of R4 by finding a matrix A so that U = Col A. b. Find a basis for U and give the dimension of U. 14. Mark each statement as true or false. Justify your answers. a. The null space of a matrix A is the set of solutions to Ax = 0. b. The null space of a 4 x 5 matrix is in R4 c. The column space of a 4 x 5 matrix is in IR4. d. The column space of the transpose of a 4 x 5 matrix is in R4. e. The null space of the transpose of a 4 x 5 matrix is in IR4. f. Let A E Maxs and b E R*. If Ax = b is consistent then Col A = IR4. g. Let A E Max5. Nul A is the kernel of the mapping x - Ax. h. Let A E M4x5. dim Nul A 2 1. 15. Let V and W be vector spaces and suppose 7: V - W is a linear transformation. Show that the range of T is a subspace of W. 16. Find all functions p(t) E Ps such that d2 de5 p t) + (3tz -1) 2P(t) + 3p(t) = 3+5 - 3t2 + 2t - 1 17. Let X = ($2 ). (3).( 56 . Find a basis for span X, 18. Mark the following statements as true or false. Justify your answers. a. If u E R* then {u} is a linearly independent set. b. The set of columns of an invertible n x n matrix is a basis for R". c. Suppose V is a vector space and V = span{v1, V2, ..., Vx]. Then {v1, V2, ..., Vx} is a basis for V. d. Suppose W is a vector space and W = span{v1, V2, ..., Vx}. Then dim W S k. . Suppose H is a vector space and {V1, V2, ..., Vx} is a linearly independent subset of H. Then {v1, V2, ..., Uk] is a basis for H. f. Suppose W is a vector space and W = span{v1, V2, ..., Vx]. Then some subset of {v1, V2, ..., VK] is a basis for W. g. If B is an echelon form of a matrix A, then the set containing the pivot columns of B is a basis for Col A. 19. Show that {t, sin(t) , cos(2t)} is a linearly independent subset of C [0, 1]. 20. Show that {1, cos(t) , cos?(t) , cost (t) , cost(t), cost(t)} is a linearly independent subset of C[0,1]