Let $S$ be the following subset of the 3-dimensional vector space $\R^3$.
\[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \]
where $\Z$ is the set of all integers.
Determine whether $S$ is a subspace of $\R^3$.

We claim that $S$ is not a subspace of $\R^3$.
If $S$ is a subspace, then $S$ is closed under scalar multiplication.
But this is not the case for $S$.

For example, consider $\mathbf{x}=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}$.
Since all entries are integers, this is an element of $S$.
Let us compute the scalar multiplication of this vector $\mathbf{x}$ and the scalar $1/2 \in \R$. We have
\[\frac{1}{2}\cdot \mathbf{x}= \frac{1}{2}\cdot\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}=\begin{bmatrix}
1/2 \\
1/2 \\
1/2
\end{bmatrix}.\]
Since $1/2$ is not an integer, the scalar multiplication $\frac{1}{2}\mathbf{x}$ is not in $S$. Therefore the subset $S$ cannot be a subspace of $R^3$.

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 […]

Subspaces of the Vector Space of All Real Valued Function on the Interval
Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not.
(a) $S=\{f(x) \in V \mid f(0)=f(1)\}$.
(b) $T=\{f(x) \in V \mid […]

Any Vector is a Linear Combination of Basis Vectors Uniquely
Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as
\[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,\]
where $c_1, c_2, c_3$ are […]

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 […]

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 $V$ […]

Every Plane Through the Origin in the Three Dimensional Space is a Subspace
Prove that every plane in the $3$-dimensional space $\R^3$ that passes through the origin is a subspace of $\R^3$.
Proof.
Each plane $P$ in $\R^3$ through the origin is given by the equation
\[ax+by+cz=0\]
for some real numbers $a, b, c$.
That is, the […]

12 Examples of Subsets that Are Not Subspaces of Vector Spaces
Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) \[S_1=\left \{\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in […]

If Vectors are Linearly Dependent, then What Happens When We Add One More Vectors?
Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent $n$-dimensional real vectors.
For any vector $\mathbf{v}_{r+1} \in \R^n$, determine whether the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly […]