Kunze Linear Algebra · Problem 2.3.7

Vector Spaces

Mathematics — Methods · subspaces-dimensions

Problem

Let

V

be the vector space of Exercise 6. Let

W_1

be the set of matrices of the form

\begin{bmatrix} x & -x \\ y & z \end{bmatrix}

and let

W_2

be the set of matrices of the form

\begin{bmatrix} a & b \\ -a & c \end{bmatrix}.

(a) Prove that

W_1

and

W_2

are subspaces of

V

. (b) Find the dimensions of

W_1

W_2

W_1 + W_2

, and

W_1 \cap W_2