\( \newcommand{\vect}[1]{\mathbf{#1}} \newcommand{\mat}[1]{\mathbf{#1}} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\Prob}{\mathbb{P}} \newcommand{\T}{^{\top}} \newcommand{\norm}[1]{\lVert #1 \rVert} \newcommand{\abs}[1]{\lvert #1 \rvert} \newcommand{\ihat}{\mkern1mu\hat{\boldsymbol{\imath}}\mkern2mu} \newcommand{\jhat}{\mkern1mu\hat{\boldsymbol{\jmath}}\mkern2mu} \newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\argmax}{\operatorname*{arg\,max}} \newcommand{\argmin}{\operatorname*{arg\,min}} \newcommand{\cdata}[1]{\class{c-data}{#1}} \newcommand{\cmodel}[1]{\class{c-model}{#1}} \newcommand{\ctruth}[1]{\class{c-truth}{#1}} \newcommand{\cgrad}[1]{\class{c-grad}{#1}} \newcommand{\closs}[1]{\class{c-loss}{#1}} \newcommand{\cparam}[1]{\class{c-param}{#1}} \newcommand{\csecond}[1]{\class{c-second}{#1}} \renewcommand{\,}{\mkern3mu} \)

M02.8 · Determinants as volume scaling

A free, illustrated module of 4 lessons and one interactive in M02 Linear Algebra I: Vectors and Matrices: The area factor; Sign, orientation and three dimensions; Zero means collapse; Determinants in practice.

M02.8 L1 · Core 4 lessons · about 100 min Live · in review

Part of M02 Linear Algebra I: Vectors and Matrices in the Mathematics track.

A determinant is not a formula to memorise. It is the number by which a matrix multiplies every area, or every volume, with a sign that says whether space has been turned over. These four lessons derive ad - bc by cutting up a parallelogram, read orientation off the columns, show that determinant zero means the map has collapsed space and cannot be undone, and explain how determinants are computed and used in practice, from elimination to probability densities.

What you need first. Matrices as linear maps, and the columns of a matrix as the places where the basis vectors land (M02.4). The dot product and projection onto a line (M02.3). The third lesson also uses linear dependence (M02.2), composition of maps (M02.5) and the inverse (M02.7); the fourth uses elimination (M02.6).

Lessons

1. The area factor M02.8.1 25 min

   A linear map multiplies the area of every shape by one fixed number. See why the number is the same for every shape, derive the formula ad - bc for a 2 by 2 matrix by cutting up a parallelogram, and check it by counting squares in code.

2. Sign, orientation and three dimensions M02.8.2 25 min

   A negative determinant means the map has turned the plane over, like a mirror. Learn to read orientation from the columns of a matrix, then move up a dimension: the 3 by 3 determinant is a signed volume factor, computed by cofactor expansion.

3. Zero means collapse M02.8.3 25 min

   A determinant of zero means the map has squashed the plane flat. Prove that this happens exactly when the columns are dependent and the matrix has no inverse, then see why determinants multiply when maps are composed, and what that says about inverses and transposes.

4. Determinants in practice M02.8.4 25 min

   How determinants are really computed: by elimination, as a product of pivots, in about n cubed over 3 steps. Then where the determinant appears in machine learning, as the factor that rescales a probability density, and why solvers avoid it.

The interactive

M02.8-X1 · Area explorer. Drag the two columns of a matrix. The image of the unit square is shaded and its signed area printed; the shading changes when orientation reverses and the cell vanishes when the columns line up. It lives in Sign, orientation and three dimensions.