Can We Reduce the Number of Vectors in a Spanning Set?

Problem 707

Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^3$. Is it possible that $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$?

As an example, consider the vectors
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
2 \\
0 \\
0
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
3 \\
0 \\
0
\end{bmatrix}.\]
Then, as we have $\mathbf{v}_2=2\mathbf{v}_1$ and $\mathbf{v}_3=\mathbf{3}\mathbf{v}_1$, we see that
the set $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of the subspace $V=\Span(\mathbf{v}_1)$, and clearly $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$.

Geometrically, $V=\Span(\mathbf{v}_1)$ is a line in $\R^3$ passing through the origin and $\mathbf{v}_1$. The vectors $\mathbf{v}_2, \mathbf{v}_3$ are also on the same line. Thus, we can omit them from the spanning set $S_1$.

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Let $V$ be a subspace of $\R^n$.
Suppose that
\[S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}\]
is a spanning set for $V$.
Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.
We give two proofs. The essential ideas behind […]

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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 $\R^n$.
Proof.
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Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set
\[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\]
still a spanning set for […]

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Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$.
Let […]

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Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.
Proof.
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Let $V$ be a vector space over a scalar field $K$.
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\[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […]

Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors
Find a basis for $\Span(S)$ where $S=
\left\{
\begin{bmatrix}
1 \\ 2 \\ 1
\end{bmatrix}
,
\begin{bmatrix}
-1 \\ -2 \\ -1
\end{bmatrix}
,
\begin{bmatrix}
2 \\ 6 \\ -2
\end{bmatrix}
,
\begin{bmatrix}
1 \\ 1 \\ 3
\end{bmatrix}
\right\}$.
Solution.
We […]

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Let $W$ be the following subset of $P_3$.
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Here $p'(x)$ is the first derivative of $p(x)$ and […]