there are 15 elements in the basis B.so dim W=15
8. What is the dimension of the vector space consisting of lower triangular 5 x 5...
9. Find the dimension of each of the following vector spaces (a) The vector space of all diagonal n xn matrices. (b) The vector space of all symmetric n x n matrices. (c) The vector space of all upper triangular n x n matrices 9. Find the dimension of each of the following vector spaces (a) The vector space of all diagonal n xn matrices. (b) The vector space of all symmetric n x n matrices. (c) The vector space...
1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all lower triangular 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, c, 2a + 3b – 3c) (which is a subspace of R4).
8. Let Maxn denote the vector space of all n x n matrices. a. Let S C Max denote the set of symmetric matrices (those satisfying AT = A). Show that S is a subspace of Mx. What is its dimension? b. Let KC Maxn denote the set of skew-symmetric matrices (those satisfying A' = -A). Show that K is a subspace of Max. What is its dimension?
8 and 11 Will h x n lower triangular matrices. Show it's a w It's a 8. Dan will represent the set of all n x n diagonal matrices. Show it's a subspace of Mr. 9. For a square matrix AE M , define the trace of A, written tr(A) to be the sum of the diagonal entries of A (i.e. if A= a) then tr(A) = 211 + a2 + ... + ann). Show that the following subset of...
(1 point) Let Ps be the vector space of all polynomials of degree at most 3, and consider the subspace 11 = {r(z) e Pal p(1) = 0} of P3 a A basis for the subspace H is { 22x+12x^2-x-1 Enter your answer as a comma separated list of polynomials. b. The dimension of His 3 (1 point) Find a basis for the space of symmetric 2 x 2-matrices If you need fewer basis elements than there are blanks provided,...
Long Answer Question LetV = M2x2, the vector space of 2 x 2 matrices with usual addition and scalar multiplication. Consider the set S = {M1, M2, M3} where M [ {].m=[5_1], 25 = [3 1] 1. (6 marks) Determine whether Sis linearly dependent/independent. 2. (2 marks) What is the dimension of Span(S)? 3. (2 marks) Is S a basis for V? 4. (2 marks) Is S a basis for the space of 2 x 2 upper triangular matrices? Please...
2. Let M2x2(R) be the vector space consisting of 2 x 2 matrices with real entries. Let W M2x2 (R) det (A) 0. Show that W is not a subspace of M2x2(R) A E
1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all symmetric 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, 2a +36) (which is a subspace of R).
1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all symmetric 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, 2a + 3b) (which is a subspace of R).
7. Consider the Theorem: Suppose A and B are two lower triangular matrices (Defined in 8 3.1), of order n. Then, the product AB is also a lower triangular matrix. Likewise for upper triangular matrices. (We say that the set of lower triangular matrices, of order n, is closed under multiplication.) Prove this theorem, for n = 3, by multiplying the following two matri- ces: a1 0 0 A bi b 0 1 0 0 and B 2 0 21...