Summary and Revision Notes

Use these notes to review Chapter 3 quickly.

Core Picture

A matrix is a rectangular table of numbers with row-column structure.

In computation, a matrix often acts as an operation:

input vector
-> matrix
-> output vector

The table shape controls whether the operation is allowed and what shape the output has.

Matrix entries are indexed by row and column.
Shape is rows by columns.
A matrix-vector product uses one dot product per row.
Matrix multiplication repeats the matrix-vector idea across columns.
Matrix multiplication is not coordinate-wise multiplication.
Matrix multiplication is order-sensitive: $AB$ and $BA$ are different questions, and one may be undefined.
The identity matrix leaves vectors unchanged.
A diagonal matrix scales coordinates separately.
The transpose swaps rows and columns.
Batches can be stored as matrices of examples by features.
A matrix can be read as a linear map.
Neural layers use weight matrices to mix input features into output features.

Notation	Read as	Meaning
$A$	matrix A	a table of numbers
$a_{ij}$	a i j	entry in row i, column j
$m \times n$	m by n	m rows and n columns
$Ax$	A times x	matrix-vector product
$AB$	A times B	matrix product
$I$	identity matrix	leaves vectors unchanged
$D$	diagonal matrix	scales coordinates separately
$A^T$	A transpose	swaps rows and columns
$Wx + b$	W x plus b	linear layer pre-activation

Expression	Requirement	Output
$Ax$	$A$ is $m \times n$ , $x$ has length $n$	length $m$
$AB$	$A$ is $m \times n$ , $B$ is $n \times p$	$m \times p$
$XW$	$X$ is batch by features, $W$ maps features to outputs	batch by outputs
$A^T$	$A$ is $m \times n$	$n \times m$

The next chapter will give these operations a picture.

A matrix can stretch, rotate, shear, or project vectors. Geometry explains what those words mean visually.