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

Ohio State University exam problems and solutions in mathematics

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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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
\end{bmatrix}\in \R^4 \quad \middle| \quad A\begin{bmatrix}
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.

  1. Find All the Eigenvalues of 4 by 4 Matrix
  2. Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
  3. Diagonalize a 2 by 2 Matrix if Diagonalizable
  4. Find an Orthonormal Basis of the Range of a Linear Transformation
  5. The Product of Two Nonsingular Matrices is Nonsingular
  6. Determine Whether Given Subsets in ℝ4 R 4 are Subspaces or Not (This page)
  7. Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
  8. Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable
  9. Idempotent Matrix and its Eigenvalues
  10. Diagonalize the 3 by 3 Matrix Whose Entries are All One
  11. Given the Characteristic Polynomial, Find the Rank of the Matrix
  12. Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$
  13. Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

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