W4-H1 - Math 2270 document PDF

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Fall 2020: MATH 2270 Week 4 HW 1

3.1 Spaces of Vectors     0 0 1 0  but not B =  . P 1. (a) Describe a subspace of M that contains A =  0 −1 0 0 (b) If a subspace of M does contain A and B, must it contains the identify matrix I ? (c) Describe a subspace of M that contains no diagonal matrices (that are nonzero). P 2. Which of the following subsets of R3 are actually subspaces? (a) All possible vectors (b1 , b2 , b3 ) with b1 = b2 . (b) All possible vectors (b1 , b2 , b3 ) with b1 = 1. (c) The vectors (b1 , b2 , b3 ) with b1 b2 b3 = 0. (d) All linear combinations of v = (1, 4, 0) and w = (2, 2, 2). (e) All vectors (b1 , b2 , b3 ) that satisfy b1 + b2 + b3 = 0. (f ) All vectors (b1 , b2 , b3 ) with b1 ≤ b2 ≤ b3 . P 3. True of False (and explain why). (a) The symmetric matrices in M (with AT = A) form a subspace. (b) The skew-symmetric matrices in M (with AT = −A) form a subspace. (c) The unsymmetric matrices in M (with AT 6= A) form a subspace. P 4. True of False (with a counterexample if false) (a) The vector b that are not in the column space C(A) form a subspace. (b) If C(A) contains only the zero vector, then A is the zero matrix. (c) The column space of 2A equals the column space of A. (d) The column space of A − I equals the column space of A.

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