Exercise 1.1.4

Question

Suppose $A$ is the 3 by 3 matrix ones $(3,3)$ of all ones. Find two independent vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ that solve $A \boldsymbol{x}=\mathbf{0}$ and $A \boldsymbol{y}=\mathbf{0}$. Write that first equation $A \boldsymbol{x}=\mathbf{0}$ (with numbers) as a combination of the columns of $A$. Why don't I ask for a third independent vector with $A z=0$?

Neil Z. · Accepted Answer

We have two solutions $x = \begin{bmatrix}0 \ 1 \ -1\end{bmatrix}$ and $y=\begin{bmatrix}1 \ 0 \ -1\end{bmatrix}$.

\begin{align*}
0	imes \begin{bmatrix}1 \ 1 \ 1\end{bmatrix} + 1 	imes \begin{bmatrix}1 \ 1 \ 1\end{bmatrix} - 1 	imes \begin{bmatrix}1 \ 1 \ 1\end{bmatrix} &= \begin{bmatrix}0 \ 0 \ 0\end{bmatrix}
\end{align*}

Any other solutions are linear combinations of the first two independent vectors $x$ and $y$.

Exercise 1.1.4

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Exercise 1.1.4

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