## Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors

## Problem 578

Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where

\[\mathbf{a}=\begin{bmatrix}

1 \\

0 \\

1 \\

0

\end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix}

1 \\

1 \\

0 \\

0

\end{bmatrix}, \quad \mathbf{c}=\begin{bmatrix}

0 \\

1 \\

-1 \\

0

\end{bmatrix}.\]

Namely,

\[V=\{\mathbf{x}\in \R^4 \mid \mathbf{a}^{\trans}\mathbf{x}=0, \mathbf{b}^{\trans}\mathbf{x}=0, \text{ and } \mathbf{c}^{\trans}\mathbf{x}=0\}.\]

**(a)** Prove that $V$ is a subspace of $\R^4$.

**(b)** Find a basis of $V$.

**(c)** Determine the dimension of $V$.