# Vector space of matrices, and subspaces

Show that the set of all real two-rowed square matrices form a vector space $X$. What is the zero vector? what is a basis? find $\dim X$. Give examples of subspaces of $X$. Do the symmetric matrices $x \in X$ form a subspace? the singular matrices?

I don't understand what the question means by two-rowed square matrices... Do they mean all $2 \times 2$ matrices ?

• Yes - this is often denoted by $M_{2 \times 2}(\mathbb{R}) = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a,b,c,d \in \mathbb{R} \right\}$. Your goal is to show that this is a vector space (where the vectors are $2 \times 2$ real matrices). – Batman Feb 17 '14 at 21:24

## 1 Answer

a) the zero vector is the 2 by 2 zero matrix. b) the basis is the set of 4 matrices each with a 1 and the rest are zero. c) dimX = 4 d) a subspace of X is the set of all 2 by 2 matrices with a(11) = 0 and a(ij) = 0. e) symmetric matrices do form a subspace. f) Singular matrices do not form a subspace because the + is not closed. Take x = id matrix and y = -id then x + y = 0 not singular.