Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$.
Then prove that $V$ is a subspace of $\R^n$.

To prove that $V=\{\mathbf{0}\}$ is a subspace of $\R^n$, we check the following subspace criteria.

Subspace Criteria
(a) The zero vector $\mathbf{0} \in \R^n$ is in $V$.
(b) If $\mathbf{x}, \mathbf{y} \in V$, then $\mathbf{x}+\mathbf{y}\in V$.
(c) If $\mathbf{x} \in V$ and $c\in \R$, then $c\mathbf{x} \in V$.

Condition (a) is clear since $V$ consists of the zero vector $\mathbf{0}$.

To check condition (b), note that the only element in $V=\{\mathbf{0}\}$ is $\mathbf{0}$. Thus if $\mathbf{x}, \mathbf{y} \in V$, then both $\mathbf{x}, \mathbf{y}$ are $\mathbf{0}$. Hence
\[\mathbf{x}+\mathbf{y} =\mathbf{0}+\mathbf{0}=\mathbf{0}\in V\]
and condition (b) is met.

To confirm condition (c), let $\mathbf{x}\in V$ and $c\in \R$. Then $\mathbf{x}=\mathbf{0}$.
We have
\[c\mathbf{x}=c\mathbf{0}=\mathbf{0}\in V\]
and condition (c) is satisfied.

Hence we have checked all the subspace criteria, and hence the subset $V=\{\mathbf{0}\}$ consisting only of the zero vector is a subspace of $\R^n$.

What’s the dimension of the zero vector space?

What’s the dimension of the subspace $V=\{\mathbf{0}\}$?

The dimension of a subspace is the number of vectors in a basis. So let us first find a basis of $V$.

Note that a basis of $V$ consists of vectors in $V$ that are linearly independent spanning set. Since $0$ is the only vector in $V$, the set $S=\{\mathbf{0}\}$ is the only possible set for a basis.

However, $S$ is not a linearly independent set since, for example, we have a nontrivial linear combination $1\cdot \mathbf{0}=\mathbf{0}$.

Therefore, the subspace $V=\{\mathbf{0}\}$ does not have a basis.
Hence the dimension of $V$ is zero.

Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis
Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient.
Let $W$ be the following subset of $P_3$.
\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\]
Here $p'(x)$ is the first derivative of $p(x)$ and […]

Every Basis of a Subspace Has the Same Number of Vectors
Let $V$ be a subspace of $\R^n$.
Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$.
Prove that every basis of $V$ consists of $k$ vectors in $V$.
Hint.
You may use the following fact:
Fact.
If […]

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 […]

Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors
Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where
\[\mathbf{a}=\begin{bmatrix}
1 \\
0 \\
1 \\
0
\end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix}
1 \\
1 […]

Vector Space of Polynomials and Coordinate Vectors
Let $P_2$ be the vector space of all polynomials of degree two or less.
Consider the subset in $P_2$
\[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\]
where
\begin{align*}
&p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\
&p_3(x)=2x^2, &p_4(x)=2x^2+x+1.
\end{align*}
(a) Use the basis […]

Dimension of the Sum of Two Subspaces
Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.
Then prove that
\[\dim(U+V) \leq \dim(U)+\dim(V).\]
Definition (The sum of subspaces).
Recall that the sum of subspaces $U$ and $V$ is
\[U+V=\{\mathbf{x}+\mathbf{y} \mid […]

Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space
Let $V$ be the following subspace of the $4$-dimensional vector space $\R^4$.
\[V:=\left\{ \quad\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \in \R^4
\quad \middle| \quad
x_1-x_2+x_3-x_4=0 \quad\right\}.\]
Find a basis of the subspace $V$ […]