Linear transformations and matrices | Chapter 3, Essence of linear algebra

Linear transformations in two dimensions are completely determined by where they send the two basis vectors, and matrices are just a compact way to store that information. The payoff is practical: once the images of i-hat and j-hat are known, the destination of any vector follows automatically—no memorizing formulas or watching the transformation happen is required.

A transformation is just an input-output rule, but linear algebra focuses on a special class: linear transformations. In 2D, linearity has a visual meaning. The origin stays fixed, and straight lines stay straight—equivalently, grid lines remain parallel and evenly spaced. Nonlinear examples fail one of these conditions: some bend lines into curves, while others keep lines straight but move the origin.

The key insight comes from how linearity interacts with grid structure. Any vector v in the plane can be written as a linear combination of the basis vectors: v = x·i-hat + y·j-hat. Because a linear transformation preserves the “same linear combination” relationship, the image of v is x times the image of i-hat plus y times the image of j-hat. So the transformation’s effect on every point in the plane is forced once the two basis vectors’ landing spots are known.

This is illustrated with a concrete example. The transformation sends i-hat to (1, −2) and j-hat to (3, 0). For v = −1·i-hat + 2·j-hat, the destination must be −1·(1, −2) + 2·(3, 0) = (5, 2). The important part isn’t that the final coordinates can be read off from a picture; it’s that the same result can be deduced from the basis-vector images alone.

To describe this numerically, the four numbers that specify where i-hat and j-hat land are packaged into a 2×2 matrix. If the matrix is

[A B C D]

the first column (A, C) is the image of i-hat and the second column (B, D) is the image of j-hat. For a vector (x, y), matrix-vector multiplication computes the same linear combination: x times the first column plus y times the second column, yielding (Ax + By, Cx + Dy). Interpreting columns as transformed basis vectors makes the multiplication feel like geometry rather than rote algebra.

The method also produces familiar transformations. A 90° counterclockwise rotation sends i-hat to (0, 1) and j-hat to (−1, 0), giving columns (0, 1) and (−1, 0). A shear keeps i-hat fixed while moving j-hat to (1, 1), so the matrix has columns (1, 0) and (1, 1).

Finally, if the two transformed basis vectors are linearly dependent—one is a scaled version of the other—the transformation collapses the entire plane onto a line (the span of those dependent vectors). Overall, matrices become a language for moving space: every matrix corresponds to a specific linear transformation, and understanding that geometric meaning sets up later topics like matrix multiplication, determinants, change of basis, and eigenvalues.