Exercise 1.1.30

Question

The linear combinations of $v=(a, b)$ and $w=(c, d)$ fill the plane unless Find four vectors $u, v, w, z$ with four components each so that their combinations $c \boldsymbol{u}+d \boldsymbol{v}+e \boldsymbol{w}+f \boldsymbol{z}$ produce all vectors $\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$ in four-dimensional space.

Prakash Anand · Accepted Answer

The combinations of $v$ and $w$ fill the plane unless $v$ and $w$ lie on the same line through $(0,0)$. Four vectors whose combinations fill 4 -dimensional space: one example is the "standard basis" $(1,0,0,0),(0,1,0,0),(0,0,1,0)$, and $(0,0,0,1)$.

Exercise 1.1.30

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Exercise 1.1.30

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