Question

In 54 though 63 (3 points each), answer A if true and B if false. 54, dim(M2×3(R))= 7 55. If V and W are finite dimensional vector spaces with dim(V) < dim(W) and T ; V → W is a linear transformation then T is injective. 56. If A is a 4 ×4 matrix whose entries consist of 14 ones and 2 zeros then det (A) 0 57. M2x2(R) is a subspace of dimension four of M3x2 (R). 58. A basis of R2la] can be extended to a basis of R3]. 59. A basis of R3r] can be contracted to a basis of R2(] 60. The spaces Rs z] and Ro are isomorphic. 61. If V and W are finite dimensional vector spaces with dim(W) < dim(V) and T : V → W is a linear transformation then Ker(T) 0 62. If V and W are finite dimensional vector spaces with dim(W)< dim(V) and T : V → W is a linear transformation then Range(T) = W 63. If uw eigenvector of A. are eigenvectors of the square matrix A then v + w is an , 19

Answer 1

Maximum we can solve four sub parts

dem (M_2 times 3 (R)) = 7 false (B) dem (M_2 times 3 (R)) lessthanorequalto min { 2, 3} implies dem (m_2 times 3 (R)) lessthanorequalto 2 55 dem (v) < dem (w) false (B) T: V rightarrow w, T is injective It is always no true for example \r\n T(x, y) = (x - y, 0, x - y) T(1, 1) = (0, 0, 0) T (2, 2) = (0, 0, 0) 56 det (A) = 0 True (A) Because there is 14 (1)^3 implies at least two Rows must contain only 1 implies two rows are Dame implies det (A) = 0 57 M_22 (R) is subspace of dem four of M_3 times 2 (R) m_2 times 2 [a_11 a_21 a_12 a_22] m_3 times [b_11 b_21 b_22 b_12 b_22 b_22] No false (B) [0 0 0 0 0 0] notelementof m_2 times 2