8.13 Column Partition of a Matrix

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Column Partition of a Matrix

Consider a matrix $A \in R^{m \times n}$ . The column partition of $A$ is a description of the matrix in terms of column vectors that construct the matrix. Let the set ${A (:, 1), A (:, 2), \dots, A (:, n)} \subseteq R^{m \times 1}$ be a collection of the $n$ separate $m \times 1$ column vectors. These vectors are organized side by side to form the rectangular array:

\begin{aligned} A & = [\begin{array}{c} A (:, 1) & A (:, 2) & \dots & A (:, n) \end{array}] \\ = {[\begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}]}_{m \times n} \end{aligned}

Below is an example of when to use column partitions.

Example

Let $\vec{b} \in R^{m}$ be a linear combination of vectors ${a_{k}}_{k = 1}^{n}$ . Write out this linear combination.

We can write our linear combination as follows.

\begin{aligned} ⟹ \vec{b} & = x_{1} {\vec{a}}_{1} + x_{2} {\vec{a}}_{2} + x_{3} {\vec{a}}_{3} + \dots + x_{n} {\vec{a}}_{n} \\ = x_{1} [\begin{array}{c} a_{11} \\ a_{21} \\ ⋮ \\ a_{m 1} \end{array}] + x_{2} [\begin{array}{c} a_{12} \\ a_{22} \\ ⋮ \\ a_{m 2} \end{array}] + x_{3} [\begin{array}{c} a_{13} \\ a_{23} \\ ⋮ \\ a_{m 3} \end{array}] \\ = x_{1} A (:, 1) + x_{2} A (:, 2) + x_{3} A (:, 3) + \dots + x_{n} A (:, n) \\ = {[\begin{array}{c} A (:, 1) & A (:, 2) & A (:, 3) & \dots & A (:, n) \end{array}]}_{m \times n} [\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ ⋮ \\ x_{n} \end{array}] \\ = A_{m \times n} \cdot {\vec{x}}_{n \times 1} \end{aligned}

This is the 1 The Matrix-Vector Multiplication Problem