32. Answer the following:
a. Let (where is the vector space of 2 by 2 matrices.) Find an example that shows U is NOT a subspace of .
The matrix where U=[1 0; 0 1] the det=1. This would not be a subspace of M22.
b. Let where is the vector space of real-valued functions defined on the interval [0, 2]. Show that is a subspace of this vector space.
1. V contains the 0 vector since
2. V is closed under scalar multiplication since u = [0,2] then cu = [0,2c]
3. V is closed under addition since ?
For question (a) you need to show that the set U of matrices with determinant 0 is not a subspace of M_22. It's not enough to show that there is a matrix, that is not in U. There are matrices in U, there are matrices outside of U. What you need to show is that there are two matrices in U whose linear combination is not in U (or, at least the sum of them is not in U). ...
Linear algebra questions are demonstrated in this solution.