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Sunday, April 14, 2013

$A \vec{x}$ as linear combination of column vectors

Matrix

$$\begin{aligned} A &= \left[\begin{array}{rrrr} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn}\\ \end{array}\right] \\ \\ &= \left[\begin{array}{rrrr} \vec{v_1} & \vec{v_2} & \cdots & \vec{v_n} \\ \end{array}\right] \end{aligned}$$

Vector

$$\begin{aligned} \vec{x} = \left[\begin{array}{r} x_1 \\ x_2 \\ \vdots \\ x_n \\ \end{array}\right] \end{aligned}$$

Product

$$\begin{aligned} A \vec{x} &= \left[\begin{array}{rrrr} x_1 \vec{v_1} + x_2 \vec{v_2} + \cdots + x_n \vec{v_n} \\ \end{array}\right] \\ &= \left[\begin{array}{rrrr} x_1 \left[\begin{array}{r} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \\ \end{array}\right] + x_2 \left[\begin{array}{r} a_{12} \\ a_{22} \\ \vdots \\ a_{m2} \\ \end{array}\right] + \cdots + x_n \left[\begin{array}{r} a_{1n} \\ a_{2n} \\ \vdots \\ a_{mn} \\ \end{array}\right] \\ \end{array}\right] \end{aligned}$$

which is an m-dimensional vector.

# posted by rot13(Unafba Pune) @ 4:25 PM

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