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Recitation 2A

Definition: If AA is an m×nm\times n matrix with columns a1,…,an\mathbf{a_1}, \ldots, \mathbf{a_n} and if x\mathbf{x} is in Rn\mathbb{R}^n, then the product of AA and x\mathbf{x}, denoted by AxAx, is [a1…an][x1⋮xn]=x1a1+…+xnan.[\mathbf{a_1} \ldots \mathbf{a_n}]\begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}=x_1\mathbf{a_1} + \ldots + x_n\mathbf{a_n}.

Example 1:

[12−10−53][437]=4[10]+3[2−5]+7[−13]=[36].\begin{bmatrix}1 & 2 & -1\\0 & -5 & 3\end{bmatrix}\begin{bmatrix}4\\3\\7\end{bmatrix}=4\begin{bmatrix}1\\0\end{bmatrix}+3\begin{bmatrix}2\\-5\end{bmatrix}+7\begin{bmatrix}-1\\3\end{bmatrix}=\begin{bmatrix}3\\ 6\end{bmatrix}.

[2−380−52][47]=4[28−5]+7[−302]=[−1332−6].\begin{bmatrix}2 & -3 \\8 & 0 \\ -5 & 2 \end{bmatrix}\begin{bmatrix}4\\7\end{bmatrix}=4\begin{bmatrix}2\\8\\-5\end{bmatrix}+7\begin{bmatrix}-3\\0\\2\end{bmatrix}=\begin{bmatrix}-13\\ 32\\-6\end{bmatrix}.

Row vector rule: If AA is an m×nm\times n matrix with rows [b1⋮bm]\begin{bmatrix}\mathbf{b_1}\\ \vdots \\ \mathbf{b_m}\end{bmatrix} and if x\mathbf{x} is in Rn\mathbb{R}^n, then the product of AA and x\mathbf{x}, denoted by AxAx, is equal to [b1⋮bm]x=[b1⋅x⋮bm⋅x].\begin{bmatrix}\mathbf{b_1}\\ \vdots \\ \mathbf{b_m}\end{bmatrix}\mathbf{x}=\begin{bmatrix}\mathbf{b_1}\cdot\mathbf{x}\\ \vdots \\ \mathbf{b_m}\cdot\mathbf{x}\end{bmatrix}.

Example 2: For v1,v2,v3\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} in Rm\mathbb{R}^m, write the linear combination 3v1−5v2+7v33\mathbf{v_1}-5\mathbf{v_2}+7\mathbf{v_3} as a matrix times a vector.

Solution: 3v1−5v2+7v3=[v1,v2,v3][3−57]=Ax.3\mathbf{v_1}-5\mathbf{v_2}+7\mathbf{v_3} = [\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}]\begin{bmatrix}3\\-5\\7\end{bmatrix}=A\mathbf{x}.