# Determine Wether 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.

### More from my site

• Hyperplane Through Origin is Subspace of 4-Dimensional Vector Space Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying $2x+3y+5z+7w=0.$ Then prove that the set $S$ is a subspace of $\R^4$. (Linear Algebra exam problem, the Ohio State […]
• Subspace of Skew-Symmetric Matrices and Its Dimension Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.) (a) Prove that the subset $W$ is a subspace of $V$. (b) Find the […]
• Maximize the Dimension of the Null Space of $A-aI$ Let $A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.$ Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
• Orthonormal Basis of Null Space and Row Space Let $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}$. (a) Find an orthonormal basis of the null space of $A$. (b) Find the rank of $A$. (c) Find an orthonormal basis of the row space of $A$. (The Ohio State University, Linear Algebra Exam […]
• Given All Eigenvalues and Eigenspaces, Compute a Matrix Product Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces $E_2=\Span\left \{\quad \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix} 1 \\ 2 \\ 1 \\ 1 […] • True or False Problems of Vector Spaces and Linear Transformations These are True or False problems. For each of the following statements, determine if it contains a wrong information or not. Let A be a 5\times 3 matrix. Then the range of A is a subspace in \R^3. The function f(x)=x^2+1 is not in the vector space C[-1,1] because […] • Quiz 8. Determine Subsets are Subspaces: Functions Taking Integer Values / Set of Skew-Symmetric Matrices (a) Let C[-1,1] be the vector space over \R of all real-valued continuous functions defined on the interval [-1, 1]. Consider the subset F of C[-1, 1] defined by \[F=\{ f(x)\in C[-1, 1] \mid f(0) \text{ is an integer}\}.$ Prove or disprove that $F$ is a subspace of […]
• Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by $W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.$ (a) Which of the following vectors are in the subset […]

### 2 Responses

1. 06/28/2017

[…] Determine Wether Given Subsets in ℝ4 R 4 are Subspaces or Not […]

2. 08/17/2017

[…] Determine Wether Given Subsets in ℝ4 R 4 are Subspaces or Not […]

##### The Product of Two Nonsingular Matrices is Nonsingular

Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix....

Close