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

linear combination problems and solutions in linear algebra

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 $V$?

 
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Solution.

Yes, in general, $S_2$ can be a spanning set.

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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