Exercise 1.1.1

Question

Give an example where a combination of three nonzero vectors in $\mathbf{R}^{4}$ is the zero vector. Then write your example in the form $A x=0$. What are the shapes of $A$ and $\boldsymbol{x}$ and $\mathbf{0}$?

Neil Z. · Accepted Answer

Consider the following example:
\begin{align*}
1	imes \begin{bmatrix}0\1\2\ 3\end{bmatrix} + 1 	imes \begin{bmatrix}1\2\-1\-3\end{bmatrix} - 1 	imes \begin{bmatrix}1\3\1\0\end{bmatrix} &= \begin{bmatrix}0\0\0\0\end{bmatrix}\
\end{align*}

If we let $$A:=\begin{bmatrix}0&1 &1 \1 &2 &3\ 2 & -1 & 1\3 & -3 & 0\end{bmatrix}, \qquad x := \begin{bmatrix}1\1\-1\end{bmatrix}$$
then we can rewrite the above example in the matrix form $A_{4	imes 3}x_{3	imes 1}=0_{4	imes 1}$.

Exercise 1.1.1

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Exercise 1.1.1

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