Linear Map - Matrices

Matrices

If V and W are finite-dimensional, and one has chosen bases in those spaces, then every linear map from V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear maps: if A is a real m × n matrix, then the rule f(x) = Ax describes a linear map Rn → Rm (see Euclidean space).

Let {v₁, ..., v_n} be a basis for V. Then every vector v in V is uniquely determined by the coefficients c₁, ..., c_n in

If f: V → W is a linear map,

which implies that the function f is entirely determined by the values of f(v₁), ..., f(v_n).

Now let {w₁, ..., w_m} be a basis for W. Then we can represent the values of each f(v_j) as

Thus, the function f is entirely determined by the values of a_ij.

If we put these values into an n × m matrix M, then we can conveniently use it to compute the value of f for any vector in V. For if we place the values of c₁, ..., c_n in an n × 1 matrix C, we have MC = the m × 1 matrix whose ith element is the coordinate of f(v) which belongs to the base w_i.

A single linear map may be represented by many matrices. This is because the values of the elements of the matrix depend on the bases that are chosen.