Fix the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let $\R^3$ be the vector space of $3 \times 1$ column vectors. Define
\[W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.\]
Prove that $W$ is a vector subspace of $\R^3$.

We verify the subspace criteria: the zero vector $\mathbf{0}$ of $\R^3$ is in $W$, and $W$ is closed under addition and scalar multiplication.

First, the zero element in $\R^3$ is $\mathbf{0}$, the $3 \times 1$ column vector whose entries are all $0$. Then clearly $\mathbf{b} \mathbf{0} = 0$, and so $\mathbf{0} \in W$.

Because, again, $\mathbf{b} \mathbf{v} = \mathbf{0}$, we have
\[\mathbf{b} ( c \mathbf{v} ) = c \mathbf{b} \mathbf{v} = c \mathbf{0} = \mathbf{0}.\]
Thus $c \mathbf{v} \in W$. These three criteria show that $W$ is a vector subspace of $\R^3$.

Comment.

We can generalize the problem with an arbitrary $1\times 3$ row vector $\mathbf{b}$.

The proof is almost identical.
(Look at the proof. We didn’t use components of the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$.)

Note that vectors $\mathbf{u}, \mathbf{v}\in \R^3$ is said to be perpendicular if
\[\mathbf{u}\cdot \mathbf{v}=\mathbf{u}^{\trans}\mathbf{v}=0.\]

Thus, the result of the problem says that for a fixed vector $\mathbf{u}\in \R^3$, the set of vectors $\mathbf{v}$ that are perpendicular to $\mathbf{u}$ is a subspace in $\R^3$.
(Note that we appy the problem to $\mathbf{b}=\mathbf{u}^{\trans}$.)

Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)
Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.
Hint.
Recall that the rank of a matrix $A$ is the dimension of the range of $A$.
The range of $A$ is spanned by the column vectors of the matrix […]

The Column Vectors of Every $3\times 5$ Matrix Are Linearly Dependent
(a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent.
(b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.
Proof.
(a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly […]

Subset of Vectors Perpendicular to Two Vectors is a Subspace
Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by
\[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\]
Prove that the subset $W$ is a subspace of […]

The Centralizer of a Matrix is a Subspace
Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define
\[W = \{ A \in V \mid AM = MA \}.\]
The set $W$ here is called the centralizer of $M$ in $V$.
Prove that $W$ is a subspace of $V$.
Proof.
First we check that the zero […]

Prove that the Center of Matrices is a Subspace
Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define
\[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\]
The set $W$ is called the center of $V$.
Prove that $W$ is a subspace […]

Subspaces of Symmetric, Skew-Symmetric Matrices
Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.
(a) The set $S$ consisting of all $n\times n$ symmetric matrices.
(b) The set $T$ consisting of […]

Determine the Values of $a$ so that $W_a$ is a Subspace
For what real values of $a$ is the set
\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\]
a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?
Solution.
The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]

Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace
Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let
\[W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}\]
be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.
Proof. […]