Matrix Representation

Linear Algebra (Matrices) · Axiom Academy

LESSON Every Linear Map Is a Matrix Apply T to the standard basis vectors, stack the images as columns, and the matrix is built — for every . 1. The Columns Are the Images of the Basis Watch where T sends the two standard basis vectors. The image slides into the first column of A , and into the second . The matrix is literally assembled out of those two output vectors. the standard matrix is the images of the basis, side by side Any input is a combination of the basis vectors. Linearity carries that same combination over to the outputs, so is the matching combination of the columns of A — and that is exactly what the matrix-vector product computes. — every vector is a weighted sum of the basis. By linearity — the weights ride along. are the columns of A , so this is . A matrix times a vector is a linear combination of its columns. The two sides agree. 3. The Columns Are the Image of the Unit Square Here is the geometric picture. The standard basis spans the unit square ; its two edges are and . Apply T and that square is carried to the parallelogram whose edges are the columns of A — so the matrix is the map's footprint made visible. Worked example — find A from scratch. Apply the shear to each basis vector. Stack the two images as columns: Check it on a test input . Directly, . Through the matrix, gives the same thing:

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