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Core Concept Map Language

Methods from Matrix representations

All methods from Matrix representations.

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[METHOD]

Construct a matrix representation

For a linear map $T:V\to W$ :

choose an ordered basis $B_V=(v_1,\ldots,v_n)$ for $V$ and an ordered basis $B_W=(w_1,\ldots,w_m)$ for $W$ ;
evaluate $T(v_j)$ for each domain basis vector;
compute the codomain coordinate vector $[T(v_j)]_{B_W}$ for each image;
place these coordinate vectors, in the same order, as the columns of the $m\times n$ matrix.

[1]:Matrix representation of a linear map
[2]:Coordinate vector
[3]:Differentiation on polynomial spaces

This page builds on Linear transformations, Rank-nullity, Coordinates and bases (Vector spaces), Kernel, image and rank, Basis and dimension (Vector spaces) and 1 more.

It leads to Seen Questions, Eigenvalues and eigenvectors (Eigenvalues), Diagonalisation (Eigenvalues), Seen Questions (Eigenvalues), Linear ODE systems (Eigenvalues) and 1 more.

Part of the thread Coordinates and linear structure, in 5 places.

A basis turns vectors and maps into columns and matrices; a good basis makes the calculation simple.

THMatrix representation of a linear mapa linear map in coordinatesALG 3 / Here

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Construct a matrix representation