Linear Algebra · Chapter 03

Matrices and transformations

From a system’s coefficient vectors to a matrix: its special shapes, its arithmetic, the map it performs, products, inverses and the transpose.

T (x) = A x

(a c b d)

(A B)_{ij} = k \sum a_{ik} b_{k j}

T (u + v) = T (u) + T (v)

A^{T}

A I = I A = A

(A B)^{T} = B^{T} A^{T}

A_{m \times n}

Lessons

Build the foundation

01What Is a Matrix?From a system’s coefficient vectors to a matrix: its special shapes, its arithmetic, the map it performs, products, inverses and the transpose.Read lesson →
02Square, Diagonal, and Identity MatricesFrom a system’s coefficient vectors to a matrix: its special shapes, its arithmetic, the map it performs, products, inverses and the transpose.Read lesson →
03Matrices and shapeA matrix is a rectangular block of numbers. For everything that follows it is best read as a row of column vectors standing side by side, because that is how a system of equations produced it.Read lesson →
04Adding and scaling matricesTwo matrices of the same shape add entry by entry, and a number scales every entry at once. It is the arithmetic of vectors, carried over to a rectangular block of numbers.Read lesson →
05A matrix times a vectorMultiplying a matrix by a vector is not a new idea. The vector supplies one weight per column, and the answer is the linear combination of the columns with those weights.Read lesson →
06Linear maps

A map takes a vector and returns a vector. It is called linear when it respects the two operations: adding first and mapping gives the same as mapping first and adding, and the same for scaling.

Read lesson →

07The matrix of a mapA linear map on all of a space is already decided by what it does to a handful of vectors. Knowing where each of them goes is enough to know where everything goes.Read lesson →

08Composing mapsDoing one linear map and then another is again a linear map, so it has a matrix of its own. That matrix is called the product of the two, and it is defined so that this is true.Read lesson →

09Computing productsEach entry of a product pairs one row of the first matrix with one column of the second: multiply the entries in pairs and add. This rule and the column-by-column definition are the same product, seen at two scales.Read lesson →

10Undoing a mapSome linear maps can be undone: there is a second map that returns every vector to where it started. Others cannot, because they send different vectors to the same place and the information is gone.Read lesson →

11Transpose and symmetric matricesTransposing a matrix turns its rows into columns: the first row becomes the first column, the second row the second column, and so on. On the page it is a reflection of the block across its main diagonal.Read lesson →

12Chapter practiceBring the chapter together: read and classify a matrix, add and scale, apply one to a vector, decide whether a map is linear, build a matrix from where the standard vectors go, multiply, invert and transpose. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Matrices and transformations

Build the foundation