W2MATH 221 ALL

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W2MATH 221 ALL 1. (1 point) Find the value of k for which the matrix A = 2 4 −5 94 − − −9 7 4 3 −k8 3 5 has rank 2. k = Correct Answers: • -14 2. (1 point) Find the rank and the nul... lity of the matrix A = 2 4 −0 0 0 1 1 1 2 0 5 1 1 −1 0 −3 −1 3 5: rank(A) = nullity(A) = rank(A)+nullity(A) = • choose • the number of columns of A • the number of rows of A Correct Answers: • 3 • 2 • the number of columns of A 3. (1 point) Find all values of x for which rank(A) = 2. A = 2 4 -1 1 0 5 2 1 x -3 3 0 -2 -8 3 5 x = Correct Answers: • -2 4. (1 point) Suppose that A is a 6 × 5 matrix which has a null space of dimension 3. The rank of A is rank(A) = Correct Answers: • 2 5. (1 point) Find a non-zero 2×2 matrix such that  −369 7 −28    =  0 0 0 0 . Correct Answers: •  −−00::111111 142857 −−00::111111 142857  6. (1 point) Determine if the set of vectors is a basis of R5. If not, determine the dimension of the subspace spanned by the vectors. 266664 -1 -1 1 -1 -1 377775 266664 10 -1 -2 1 377775 266664 -2 -1 2 -1 -2 377775 266664 2 -1 -2 -1 2 377775 266664 -1 -2 10 -1 377775 The dimension of the subspace spanned by the vectors is Correct Answers: • 3 7. (1 point) Let B = 2664 3 4 6 −2 1 −8 −3 0 −8 −2 −2 −3 3775 : (a) Find the reduced row echelon form of the matrix B. rref(B) = 2664 3775 (b) How many pivot columns does B have? (c) Do the vectors in the set 8>><>>: 2664 3 −2 −3 −2 3775 ; 2664 410 −2 3775 ; 2664 6 −8 −8 −3 3775 9>>=>>; span R4? Be sure you can explain and justify your answer. • choose • the vectors span Rˆ4 • the vectors do not span Rˆ4 (d) Are the vectors in the set 8>><>>: 2664 3 −2 −3 −2 3775 ; 2664 410 −2 3775 ; 2664 6 −8 −8 −3 3775 9>>=>>; linearly independent? Be sure you can explain and justify your answer. • choose • linearly dependent • linearly independent Correct Answers: 2664 1 0 0 0 1 0 0 0 1 0 0 0 3775 • 3 • the vectors do not span Rˆ4 • linearly independent 8. (1 point) Are the following statements true or false? ? 1. If T : R3 ! R9 is a linear transformation, then range (T) (also known as the image of T) is a subspace of R9. ? 2. The sum of two subspaces of Rn forms another subspace of Rn. The sum of V and W means the set of all vectors ~v +~w where ~v is an element of V and ~w is an element of W. ? 3. The intersection of two subspaces of Rn forms another subspace of Rn. ? 4. If u and v are in a subspace S, then every point on the line connecting u and v is also in S. [The line is the set of vectors you can form as tu+(1−t)v for different values of t] Correct Answers: • T • T • T • T 9. (1 point) If A is an n × n matrix and b 6= 0 in Rn, then consider the set of solutions to Ax = b. Select true or false for each statement. ? 1. This set is closed under scalar multiplications ? 2. The set contains the zero vector ? 3. This set is closed under vector addition ? 4. This set is a subspace Correct Answers: • FALSE • FALSE • FALSE • FALSE 10. (1 point) Find the determinant of the matrix M = 2664 −2 0 0 −1 −1 0 3 0 0 −2 0 2 0 2 2 0 3775 : det(M) = . Correct Answers: • -2*3*2*2--1*-1*-2*2 11. (1 point) Find k such that the following matrix M is not invertible (singular). M = 2 4 10−3 2 14 +1 0 k 2 10 −4 3 5 k = Correct Answers: • -8 12. (1 point) Given the matrix A = a+018 a− −018 find all values of a that make det(A) = 0. Give your answer as a comma-separated list. Values of a: . Correct Answers: • -18, 18 13. (1 point) Evaluate the following 4 × 4 determinant. Use the properties of determinants to your advantage. [Show More]

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