Matrix-Vector Multiplication

Definition

Matrix-vector multiplication applies a matrix to a vector.

Basic form:

A * v = output vector

The matrix A contains the transformation rule.

The vector v is the input.

Matrix-vector multiplication transforms a vector.

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Shape Requirement

The number of matrix columns must match the number of vector components.

Example:

A is 2 x 3
v has 3 components

Then:

A * v is valid

The output has one component for each matrix row.

So:

2 x 3 matrix * 3-component vector = 2-component vector

Matrix columns must match vector length.

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A Simple Example

Let:

A =
[2 0]
[0 3]

and:

v = [4, 5]

The result is:

A * v = [8, 15]

Because:

first output  = 2*4 + 0*5 = 8
second output = 0*4 + 3*5 = 15

Each output component comes from one row of the matrix.

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Rows Act Like Dot Products

Each row of the matrix is dot-multiplied with the input vector.

Example:

A =
[2 0]
[0 3]

v = [4, 5]

Row 1:

dot([2, 0], [4, 5]) = 8

Row 2:

dot([0, 3], [4, 5]) = 15

Output:

[8, 15]

Matrix-vector multiplication is row-by-vector dot products.

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Component Mixing Example

Let:

A =
[1 1]
[0 1]

and:

v = [3, 4]

Compute:

first output  = 1*3 + 1*4 = 7
second output = 0*3 + 1*4 = 4

Result:

[7, 4]

The first output component mixed both input components.

Matrix rows define how input components combine.

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Output Dimension

The output dimension equals the number of matrix rows.

Example:

3 x 2 matrix * 2-component vector = 3-component vector

Why?

Because the matrix has three rows.

Each row produces one output component.

Rows determine output size. Columns determine required input size.

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Why It Matters

Matrix-vector multiplication gives a precise way to transform vector data.

It can represent:

• scaling
• mixing
• projection
• coordinate changes
• feature transformations

This is one of the most important operations in linear algebra.

Matrix-vector multiplication is structured vector computation.

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Final Summary

• Matrix-vector multiplication applies a matrix to a vector.
• Matrix columns must match vector length.
• Each matrix row produces one output component.
• Each output component is a dot product.
• Rows determine output dimension.
• The operation can scale, mix, or transform vector data.