## 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*)

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