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M02.4 · Matrices as linear maps

A free, illustrated module of 4 lessons and one interactive in M02 Linear Algebra I: Vectors and Matrices: What a linear map is; Columns are where the basis vectors land; Rotation and shear as matrices; Maps between dimensions.

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

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

A matrix is not a table of numbers to memorise rules for. It is a machine that moves every point of space at once, and it does so in the one way that keeps a grid a grid. These four lessons build that picture, show that the columns of a matrix are simply where the basis vectors land, and use it to read rotations, shears, projections and maps between dimensions straight off the numbers.

What you need first. Vectors as arrows and as lists of numbers, adding vectors and scaling them (M02.1), and linear combinations (M02.2). Dot products (M02.3) help in the last lesson but are not required.

Lessons

1. What a linear map is M02.4.1 25 min

   A linear map moves every point of the plane at once while keeping grid lines straight, parallel and evenly spaced. Learn the two rules behind that picture, test formulas against them, and see why two arrows are enough to pin a linear map down.

2. Columns are where the basis vectors land M02.4.2 25 min

   A matrix is a record of where the basis vectors land, written as columns. Learn to multiply a matrix by a vector as a combination of its columns, see why that equals the row-by-row rule, and prove that matrices and linear maps are the same thing.

3. Rotation and shear as matrices M02.4.3 25 min

   Build the matrix of any linear motion of the plane by asking where the two basis vectors land: scaling, rotation by any angle, shear, reflection and projection. Derive the rotation matrix from the unit circle, and see which maps keep lengths and which keep areas.

4. Maps between dimensions M02.4.4 25 min

   A matrix need not be square. An m-by-n matrix takes n numbers in and gives m out: one column per input direction, each with m entries. See a 2-by-3 matrix squash space onto a plane, a 3-by-2 matrix place a plane inside space, and why a neural-network layer’s weights have the shape (outputs, inputs).

The interactive

M02.4-X1 · Matrix playground. Drag the two basis vectors and watch the whole plane follow. The matrix, the area factor and the image of any vector update as you move. It lives in Columns are where the basis vectors land.