Question

Let V be the vector space of 2× 2 matrices over R. Check whether the subsets
W1 = { (a,1) ,(0,-a) | a ∈ R} and W2 = { (a,-a), (0,b)| a,b∈ R}
are subspaces over R. For those sets which are subspaces, find their dimension and a
basis over R.

Answer 1

Answer on Question #65691 – Math – Linear Algebra

Question

Let $V$ be the vector space of $2 \times 2$ matrices over $R$. Check whether the subsets

$W1 = \{(a, 1), (0, -a) \mid a \in R\}$ and $W2 = \{(a, -a), (0, b) \mid a, b \in R\}$ are subspaces over $R$. For those sets which are subspaces, find their dimension and a basis over $R$.

Solution

We use the criterion subspace linear space.

Let $A = \begin{pmatrix} a & 1 \\ 0 & -a \end{pmatrix} \in W_1$ and $B = \begin{pmatrix} b & 1 \\ 0 & -b \end{pmatrix} \in W_1$.

Because $A + B = \begin{pmatrix} a + b & 2 \\ 0 & -a - b \end{pmatrix} \notin W_1$, then $W_1$ is not a subspace.

Let $A = \begin{pmatrix} a & -a \\ 0 & b \end{pmatrix} \in W_2$, $B = \begin{pmatrix} c & -c \\ 0 & d \end{pmatrix} \in W_2$ and $\alpha, \beta \in \mathbb{R}$.

Because $\alpha A + \beta B = \begin{pmatrix} \alpha a + \beta c & -(\alpha a + \beta c) \\ 0 & \alpha b + \beta d \end{pmatrix} \in W_2$, then $W_2$ is a subspace.

Using the definitions of basis and dimension one gets that the set of vectors $\left\{\begin{pmatrix} 1 & -1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\right\}$ is the basis of $W_2$ and $\dim_{\mathbb{R}} W_2 = 2$.

**Answer**: $W_1$ is not a subspace; $W_2$ is a subspace; $\left\{\begin{pmatrix} 1 & -1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\right\}$ is the basis of $W_2$; $\dim_{\mathbb{R}} W_2 = 2$.

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