1 The Matrix-Vector Multiplication Problem

Types: N/A
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Generalizations: 2A The Nonsingular Linear-Systems Problem

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Problem 1: The Matrix-Vector Multiplication Problem

Let $m, n \in N$ . Let $A \in R^{m \times n}$ be a given matrix and $x \in R^{n}$ be a given vector. Then the matrix-vector multiplication problem is to find an unknown vector $b \in R^{m}$ such that

A \cdot \vec{x} = \vec{b}

[A]_{\underset{r o w s}{m} \times \underset{c o l u m n s}{n}} \cdot [x]_{n \times 1} = [b]_{m \times 1}

When doing matrix-vector multiplication, the inner dimensions must agree! This is a "forward problem." Let's define the function

f : \underset{domain}{\underset{⏟}{R^{n}}} \to \underset{codomain}{\underset{⏟}{R^{m}}}

given by

f (\vec{x}) = A \cdot \vec{x} .

The range of $f$ is contained in $R^{m}$ but may not be equal to $R^{m}$ depending on the entries in the columns of matrix $A$ .