Does an Extra Vector Change the Span?

Problem 706

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 $V$? If so, prove it. Otherwise, give a counterexample.

Proof.

We prove that $S_2$ is also a spanning set for $V$, that is, we prove that
$\Span(S_2)=V.$

Prove $\Span(S_2) \subset V$

We first show that $\Span(S_2)$ is contained in $V$. Let $\mathbf{x}$ be an element in $\Span(S_2)$. Then there exist scalars $c_1, c_2, c_3, c_4$ such that
$\mathbf{x}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3 \mathbf{v}_3 + c_4 \mathbf{v}_4.$ Since $\Span(S_1)=V$, we know that $c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3 \mathbf{v}_3$ is a vector in $V$. As $\mathbf{v}_4\in V$, we have $c_4\mathbf{v}_4 \in V$.
Since $V$ is a vector space, the sum of two elements in $V$ is in $V$.
So, $\mathbf{x}=(c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3 \mathbf{v}_3) + (c_4 \mathbf{v}_4) \in V.$ This proves that $\Span(S_2) \subset V$.

Prove $\Span(S_2) \supset V$

Note that since $S_1$ is a spanning set for $V$, every element of $S_1$ can be written as a linear combination of the vectors $\mathbf{v}_1, \mathbf{v}_2$, and $\mathbf{v}_3$.
That is, for any $\mathbf{v}\in V$, there exist scalars $c_1, c_2, c_3$ such that
$\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3.$ Observe that this can be written as follows.
$\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3+0\mathbf{v}_4.$ This tells us that $\mathbf{v}$ is a linear combination of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$, and $\mathbf{v}_4$.
Hence, any vector in $V$ can be written as a linear combination of the vectors in $S_2$.
Thus, $V\subset \Span(S_2)$.

Putting these inclusion together yields that $V=\Span(S_2)$, and hence $S_2$ is a spanning set for $V$.

More from my site

• Linear Independent Vectors and the Vector Space Spanned By Them 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 […]
• The Subspace of Linear Combinations whose Sums of Coefficients are zero Let $V$ be a vector space over a scalar field $K$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset $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 } […] • 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 […]
• 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 […]
• Can We Reduce the Number of Vectors in a Spanning Set? 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 $V$?   Solution. Yes, in general, $S_2$ can be a spanning set. As an […]

Close