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Exercise 3.1.7

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It’s enough to check the following matrix multiplication is right. Let ${u_{1}, u_{2}, \dots, u_{n}}$ and ${v_{1}, v_{2}, \dots, v_{n}}$ be the row and column vectors of $A$ respectly.

For row operations:

\begin{matrix} i - th \\ j - th \end{matrix} (\begin{matrix} ⋱ \\ 1 \\ ⋱ \\ 1 \\ ⋱ \end{matrix}) A = (\begin{matrix} ⋱ \\ − & u_{j} & − \\ ⋱ \\ − & u_{i} & − \\ ⋱ \end{matrix})

\begin{matrix} i - th \end{matrix} (\begin{matrix} ⋱ \\ 1 \\ c \\ 1 \\ ⋱ \end{matrix}) A = (\begin{matrix} ⋱ \\ ⋱ \\ − & c u_{i} & − \\ ⋱ \\ ⋱ \end{matrix})

\begin{matrix} i - th \\ j - th \end{matrix} (\begin{matrix} ⋱ \\ 1 \\ ⋱ \\ c & 1 \\ ⋱ \end{matrix}) A = (\begin{matrix} ⋱ \\ ⋱ \\ ⋱ \\ − & c u_{i} & + & u_{j} & − \\ ⋱ \end{matrix})

For column operations:

A \overset{\begin{matrix} i & j \\ - th & - th \end{matrix}}{(\begin{matrix} ⋱ \\ 1 \\ ⋱ \\ 1 \\ ⋱ \end{matrix})} = (\begin{matrix} ⋱ \\ | & | \\ v_{j} & ⋱ & v_{i} \\ | & | \\ ⋱ \end{matrix})

A \overset{\begin{matrix} i \\ - th \end{matrix}}{(\begin{matrix} ⋱ \\ 1 \\ c \\ 1 \\ ⋱ \end{matrix})} = (\begin{matrix} ⋱ \\ | \\ c v_{i} \\ | & ⋱ \\ ⋱ \end{matrix})

A \overset{\begin{matrix} i & j \\ - th & - th \end{matrix}}{(\begin{matrix} ⋱ \\ 1 & c \\ ⋱ \\ 1 \\ ⋱ \end{matrix})} = (\begin{matrix} ⋱ & | \\ ⋱ & c v_{i} \\ ⋱ & + \\ v_{j} \\ | & ⋱ \end{matrix})

jephianlin

2011-06-27 00:00

Exercise 3.1.7

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