Exercise 1.1.3 (Practice with subscripts)

Question

The vectors $\boldsymbol{a}_{1}, \boldsymbol{a}_{2}, \ldots, \boldsymbol{a}_{n}$ are in $m$-dimensional space $\mathbf{R}^{m}$, and a combination $c_{1} \boldsymbol{a}_{1}+\cdots+c_{n} \boldsymbol{a}_{n}$ is the zero vector. That statement is at the vector level.
\begin{enumerate}
\item Write that statement at the matrix level. Use the matrix $A$ with the $\boldsymbol{a}$ 's in its columns and use the column vector $c=\left(c_{1}, \ldots, c_{n}\right)$.
\item Write that statement at the scalar level. Use subscripts and sigma notation to add up numbers. The column vector $\boldsymbol{a}_{j}$ has components $a_{1 j}, a_{2 j}, \ldots, a_{m j}$.
\end{enumerate}

Neil Z. · Accepted Answer

\begin{enumerate}
\item $Ac = 0$, where $A = \begin{bmatrix}a_1 & a_2 & \dots & a_n\end{bmatrix}_{m 	imes n}$, and $c = \begin{bmatrix}c_1 \ c_2\ \vdots \ c_n\end{bmatrix}_{n 	imes 1}$
\item $\sum^n_{j=1} a_{ij}c_j = 0$ for $i = 1, 2, \dots, m$
\end{enumerate}

Exercise 1.1.3 (Practice with subscripts)

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Exercise 1.1.3 (Practice with subscripts)

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