10.4 Row-Vector-Matrix Multiplication via Linear Combinations

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 10.4.1 Example of Row Vector Matrix Multiplication via Linear Combinations
Constructions: N/A
Generalizations: N/A

Properties: 10.7 Properties of Row-Vector-Matrix Multiplication
Sufficiencies: N/A
Questions: N/A

Row-Vector-Matrix Multiplication via Linear Combinations

Let matrix $A \in R^{m \times n}$ and 3.1 Column Vector $\vec{x} \in R^{m \times 1}$ . Then, the linear combination version of the row-vector-matrix product ${\vec{x}}^{T} A = \vec{b} \in R^{1 \times n}$ is given by the linear combination

\begin{aligned} {\vec{x}}^{T} A & = x_{1} [\begin{array}{c} a_{11} & a_{12} & \dots & a_{1 n} \end{array}] \\ + x_{2} [\begin{array}{c} a_{21} & a_{22} & \dots & a_{2 n} \end{array}] \\ ⋮ \\ + x_{m} [\begin{array}{c} a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}] \\ = \vec{b} \end{aligned}

Using summation and 8.11 Colon Notation, we write

{\vec{x}}^{T} A = x_{1} A (1, :) + x_{2} A (2, :) + \dots + x_{m} A (m, :) = \sum_{i = 1}^{m} x_{i} A (i, :) = \vec{b}

Remark. When the modeling matrix shows up on the right, we cut the matrix into rows.

\begin{aligned} \underset{algebraic worker}{{\vec{x}}^{T}} \cdot \underset{modeling matrix}{A} & = \vec{b} \\ [\vec{x}]_{1 \times m}^{T} [A]_{m \times n} & = [\begin{array}{ccc} x_{1} & x_{2} & \dots & x_{m} \end{array}]_{1 \times m} {[\begin{array}{c} A (1, :) \\ A (2, :) \\ ⋮ \\ A (m, :) \end{array}]}_{m \times 1} \\ ⟹ {\vec{x}}^{T} \cdot A & = x_{1} [A (1, :)] + x_{2} [A (2, :)] + \dots + [x_{m} A (m, :)] \\ ⟹ {\vec{x}}^{T} A & = x_{1} {[\begin{array}{c} a_{11} & a_{12} & \dots & a_{1 n} \end{array}]}_{1 \times n} \\ + x_{2} {[\begin{array}{c} a_{21} & a_{22} & \dots & a_{2 n} \end{array}]}_{1 \times n} \\ + x_{m} {[\begin{array}{c} a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}]}_{1 \times n} \end{aligned}

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Remark. Notice in the linear combination definition, we chunk the data into vector-sized pieces.
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Remark. Notice that we define row-vector-matrix multiplication as the transpose of a 3.1 Column Vector. It's defined as

\begin{aligned} [{\vec{x}}^{T}]_{1 \times m} [A]_{m \times n} & [\vec{x}] \in R^{m \times 1} \end{aligned}

rather than

\begin{aligned} [\vec{x}] \cdot [A] & [\vec{x}] \in R^{1 \times m} \end{aligned}

because of something we are going study something in the future called quadratic form:

{\vec{y}}^{T} k \vec{x}

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Remark. Notice in the linear combination definition, we chunk the data into vector-sized pieces. ⧫

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Remark. Notice in the linear combination definition, we chunk the data into vector-sized pieces.
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