(a) Show that if 𝐯 is a linear combination of 𝐯1, …, 𝐯m, and each 𝐯j is a linear combination of

### Part (a)u000Au000A**Step 1:** u000AAssume $u005Cmathbf{v}$ is a linear combination of $u005Cmathbf{v}_1, u005Cldots

(a) Show that if 𝐯 is a linear combination of 𝐯1, …, 𝐯m, and...

(a) Show that if $\mathbf{v}$ is a linear combination of $\mathbf{v}_1, \ldots, \mathbf{v}_m$, and each $\mathbf{v}_j$ is a linear combination of $\mathbf{w}_1, \ldots, \mathbf{w}_n$, then $\mathbf{v}$ is a linear combination of $\mathbf{w}_1, \ldots, \mathbf{w}_n$. (b) Suppose $\mathbf{v}_1, \ldots, \mathbf{v}_m$ span $V$. Let $\mathbf{w}_1, \ldots, \mathbf{w}_m \in V$ be any other elements. Suppose that each $\mathbf{v}_i$ can be written as a linear combination of $\mathbf{w}_1, \ldots, \mathbf{w}_m$. Prove that $\mathbf{w}_1, \ldots, \mathbf{w}_m$ also $\operatorname{span} V$.

(a) Show that if $\mathbf{v}$ is a linear combination of $\mathbf{v}_1, \ldots, \mathbf{v}_m$, and each $\mathbf{v}_j$ is a linear combination of $\mathbf{w}_1, \ldots, \mathbf{w}_n$, then $\mathbf{v}$ is a linear combination of $\mathbf{w}_1, \ldots, \mathbf{w}_n$.
(b) Suppose $\mathbf{v}_1, \ldots, \mathbf{v}_m$ span $V$. Let $\mathbf{w}_1, \ldots, \mathbf{w}_m \in V$ be any other elements. Suppose that each $\mathbf{v}_i$ can be written as a linear combination of $\mathbf{w}_1, \ldots, \mathbf{w}_m$. Prove that $\mathbf{w}_1, \ldots, \mathbf{w}_m$ also $\operatorname{span} V$.

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