Nonsquare matrices as transformations between dimensions | Chapter 8, Essence of linear algebra

Non-square matrices aren’t a special case—they’re the standard way to encode linear transformations between spaces of different dimensions. A 2D-to-3D linear map can be represented by a 3x2 matrix: the two columns describe where the 2D basis vectors land, and the three rows record those landing spots using three coordinates. Geometrically, that means all outputs lie in a 2D column space: a plane through the origin inside 3D space. Because the matrix has full rank (the column space has the same dimension as the input), it genuinely captures a two-dimensional input being mapped into a three-dimensional output without collapsing the input’s dimensionality.

The key idea is that matrix encoding still works the same way: find the images of the input basis vectors, then stack their coordinate descriptions as columns. In the example given, the transformation sends i-hat to the coordinates (2, −1, −2) and j-hat to (0, 1, 1). Writing those as columns produces a 3x2 matrix, and the “3” in the row count signals that each basis vector’s image needs three coordinates to specify its location in the output space. Meanwhile, the “2” in the column count signals there are two independent input directions—so the map is naturally interpreted as taking 2 dimensions and producing points in 3 dimensions.

The same logic flips cleanly for other shapes. A 2x3 matrix indicates a transformation from 3D down to 2D: three columns correspond to three input basis vectors, while only two rows mean each landing spot is described using just two coordinates. That dimensional reduction can feel unintuitive because multiple distinct 3D directions must “squeeze” into a 2D plane, so the mapping can’t preserve all degrees of freedom. The transcript emphasizes that the discomfort is part of the geometry: you’re mapping a higher-dimensional space onto a lower-dimensional one.

There’s also a 2D-to-1D case, encoded by a 1x2 matrix. Here the output is a number (points on the real line), and linearity is visualized by how evenly spaced points on a line of dots remain evenly spaced after mapping onto the number line. Each column of a 1x2 matrix contains a single entry because each 2D basis vector’s image is just one coordinate on the line. The discussion notes that this particular kind of transformation connects closely to the dot product, setting up the next chapter.

Overall, the takeaway is a practical geometric dictionary: the number of columns tells how many input dimensions you start with, and the number of rows tells how many coordinates you need to describe the outputs. Once that’s internalized, non-square matrices become a direct way to reason about linear maps between dimensions—whether expanding into a higher-dimensional space, compressing into a lower-dimensional one, or projecting down to a line.