Q3- Show that the the set of upper triangular matrices of order 2 is a subspace...
Let U be the set of all 2x2 upper triangular matrices with real entries show that B-{[6] [8]} is a linearly indepandewe set mo Explain why B is not a basis for U Include one more matry in B so that this becames a basis for U
Let W be the set of singular (noninvertible) matrices
of order 2. Show that W is not a subspace of
M2×2 with the standard matrix operations.
Q1: Let W be the set of singular (noninvertible) matrices of order 2. Show that W is not a subspace of M2x2 with the standard matrix operations.-
Verify the following properties, using any distinct, invertible
A, B, 4×4 upper triangular matrices of your choice:
3. (0.5 marks each) Verify the following properties, using any distinct, invertible A, B, 4 x 4 upper triangular matrices of your choice: (a) The inverse of an upper triangular matrix is upper triangular; (b) (AB)- B-1A-1 (e) trace(AB) trace(BA); (d) det(AB) det (BA) example of matrices A, B such that det(AB) det(BA) (BONUS 1 mark) Give an
3. (0.5 marks each) Verify...
(1 point) (a) If S is the subspace of M3(R) consisting of all lower triangular matrices, then dim S = (b) If S is the subspace of M4(R) consisting of all skew-symmetric matrices, then dim S =
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...
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...
LO 2a 4) Let V be the set of diagonal 2x2 matrices of the form la ). Determine whether or not this set is a subspace of the set of all real-valued 2x2 matrices, M22, with standard matrix addition and scalar multiplication. Justify your answer.
[B] Let W be the subspace of M22 given in problem [A] . (B.1) Show that the following set forms a basis for W: S = -5 (B.2) Obtain the coordinate vector for A = 3 relative to S. That is, find (A)s. -8 Show work!
[B] Let W be the subspace of M22 given in problem [A] . (B.1) Show that the following set forms a basis for W: S = -5 (B.2) Obtain the coordinate vector for A...
3. Let Un (R) be the subgroup of GLn(R) consisting of upper triangular matrices and let Dn(R) be the subgroup of GLn(R) consisting of diagonal matrices. (a) Show that \ : Un(R) + Dn(R), A + diag(a11, ..., ann) is a homomorphism of groups. Find the kernel and image of y. (b) Let Zn(R) = {aIn : a E R*} be the subgroup of Dn(R) consisting of scalar matrices. Determine x-1(Zn(R)). Justify your answers.
3. (a) Show the set of all matrices of the form х A у x + y + z 2 is a subspace of the vectors space M2(R) of all 2 x 2 matrices with entries in R. (b) Find a basis for this subsace and prove that it is a basis. (c) What is the dimension of this subspace?