need help with question no 3rd and 4th
abcd123
2.docx1. (5 points) Let T : Mnn(R) → R defined by T(A) = rank(A). Is T a linear transformation? If it is, show it; otherwise, provide an example to disprove it.
Let V = Mn×n(R) be the space of n×n matrices over R. Then the mapping
tr : V → R, tr(A) =
(The trace) is linear.
Similarly taking transposes of matrices is linear; i.e. we have a linear mapping
Where the transpose AT of A is the n × m matrix
AT = [ˆaij], where ˆaij = aji.
2. (5 points) Let W be a subspace of a finite-dimensional vector space V. Prove that dimW = dimV if and only if W = V
Proof: Pick a basis for W. It has dim V vectors, since W and V have the same dimension. But this list is also an independent list of vectors in V, so by the extension lemma it can be extended to a basis of V. But since the list already has the right length, we do not need to add in any vectors, i.e., it is already a basis for V
