# Determine Whether Given Subsets in $\R^4$ are Subspaces or Not

## Problem 480

(a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying
$2x+4y+3z+7w+1=0.$ Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(b) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying
$2x+4y+3z+7w=0.$ Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(These two problems look similar but note that the equations are different.)

(The Ohio State University, Linear Algebra Final Exam Problem)

## Solution.

### (a) $2x+4y+3z+7w+1=0$

We claim that $S$ is not a subspace of $\R^4$.
If $S$ is a subspace of $\R^4$, then the zero vector $\mathbf{0}=\begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ in $\R^4$ must lie in $S$.

However, the zero vector $\mathbf{0}$ does not satisfy the equation
$2x+4y+3z+7w+1=0.$

So $\mathbf{0} \not \in S$, and we conclude that $S$ is not subspace of $\R^4$.

### (b) $2x+4y+3z+7w=0$

In a set theoretical notation, we have
$S=\left\{\, \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}\in \R^4 \quad \middle| \quad 2x+4y+3z+7w=0 \,\right\}.$

Let $A$ be the $1\times 4$ matrix defined by
$A=\begin{bmatrix} 2 & 4 & 3 & 7 \end{bmatrix}.$ Then the equation $2x+4y+3z+7w=0$ can be written as
$A\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}=0.$

So we have
\begin{align*}
S&=\left\{\, \begin{bmatrix}
x \\
y \\
z \\
w
x \\
y \\
z \\
w
\end{bmatrix}=0 \,\right\}\\
&=\calN(A),
\end{align*}
the null space of $A$.

Recall that the null space of a matrix is always a subspace.
Hence the subset $S$ is a subspace of $\R^4$ as it is the null space of the matrix $A$.

## Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

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