Exercise 2.5.5

Question

Let vectors $\mathbf{u}$, $\mathbf{v}$, $\mathbf{w}$ be a basis in $V$. Show that $\mathbf{u + v + w}$, $\mathbf{u + v}$, $\mathbf{w}$ is also a basis in $V$.

Jing Huang · Accepted Answer

extbf{Solution} For any vector $\mathbf{x} \in V$, suppose $\mathbf{x} = x_1\mathbf{u} + x_2 \mathbf{v} + x_3 \mathbf{w}$. Then it is easy to figure out the $\mathbf{x} = x_1 (\mathbf{u + v + w}) + (x_2 - x_1) (\mathbf{v + w}) + (x_3 - x_2 - x_1)\mathbf{w}$. Thus $\mathbf{u + v + w}$, $\mathbf{u + v}$, $\mathbf{w}$ is also a basis in $V$.

Exercise 2.5.5

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Exercise 2.5.5

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