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.
Homework Answers
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We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be
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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....
Let V and W be finite dimensional vector spaces over R and T:V + W be linear. Let V be a subspace of V and Wo = T(V). (Select ALL that are TRUE) If T is surjective then Vo = {v EV : there is w E Wo such that T(v) = w} If T is injective then dim(VO) = dim(W). dim(ker(T) n Vo) = dim(VO) - dim(Wo).
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