
Problem 12 Show that the following set is a subspace of M3x3, the space of all...
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be
the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l
A^T=-A}.
(a) Show that W is a subspace of M2x2(R)
(b) Find a basis for W and determine dim(W).
(c) Suppose T: M2x2(R) is a linear transformation given by
T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You
do not need to verify that T is linear.
3. (17 points)...
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?
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?
Let n EN Consider the set of n x n symmetric matrices over R with the usual addition and multiplication by a scalar (1.1) Show that this set with the given operations is a vector subspace of Man (6) (12) What is the dimension of this vector subspace? (1.3) Find a basis for the vector space of 2 x 2 symmetric matrices (6) (16)
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...
please show the details thanks?
6. Recall that the set of all 3 × 3 symmetric matrices is a vector space. Compute its dimension. (Yo must find a candidate basis and then verify that it is a basis.)
Being F the subset of of the hemi-symmetric matrices (
such as ).
i) Show that F is a
subspace of .
ii) Determine the dimension of
F.
iii) Determine the base of
F.
iv) Being the application that corresponds
to each matrix
of F the vector of .
Determine the matrix that represents T
regarding the base of the previous question (iii) and the canonical
base of .
v) Determine if T is
injective.
vi) Determine if T is
surjective....
Why does this show that H is a subspace of R3? O A. The vector v spans both H and R3, making H a subspace of R3. OB. The span of any subset of R3 is equal to R3, which makes it a vector space. OC. It shows that H is closed under scalar multiplication, which is all that is required for a subset to be a vector space. OD. For any set of vectors in R3, the span of...
7. V={[)a620) a vector space! Draw the vector space? Draw the graph and explain why or why not? I. Verify the axiom for polynomial. p(x) = 2t' +31° +1+1 9(x) = 4r +57 +31 + 2 8. p(t)+9(1) € P. 9. p(t)+q(t) = f(t)+p(1) 10. cp(1) EP A subspace of a vector V is a subset H that satisfies what three conditions? 12. Is 0 a subspace of R" 13. Let V, V, E V; show H = span{v. v)...
1. Verify that the set V, consisting of all scalar multiples of (1,-1, -2) is a subspace of R. 2. Let V, be the set of all 2 x 3 matrices. Verify that V, is a vector space. 3. Let A=(1-11) Let V, be the set of vectors x € R such that Ax = 0. Verify that V, is a subspace of R. Compare V, with V.