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 $R^3$.

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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 $P_3$.
\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\]
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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.
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(b) $T=\{f(x) \in V \mid […]

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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
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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 […]

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Let $V$ be the following subspace of the $4$-dimensional vector space $\R^4$.
\[V:=\left\{ \quad\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \in \R^4
\quad \middle| \quad
x_1-x_2+x_3-x_4=0 \quad\right\}.\]
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Proof.
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\[ax+by+cz=0\]
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That is, the […]

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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.
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x_1 \\
x_2 \\
x_3
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in […]