MATH.18:5.3 - Compare linear maps with matrices, then add a length question

Additional structure – finite-dimensional real vector spaces. In one account, each space comes with a specified ordered basis, and maps between spaces are all linear maps. In the other, an object is a nonnegative integer n and a map n to m is an m-by-n real matrix. Maps compose by matrix multiplication.

For a space V with basis (b1,...,bn), let c_V:V -> R^n send a vector to its coefficients in that basis. Interpret f:V -> W by the matrix of c_W∘f∘c_V^(-1). Composition agrees because the adjacent coordinate conversion and its inverse cancel. Identity maps become identity matrices.

The return takes n to R^n with its standard basis and a matrix to its linear map. The round trip on matrices is literal. The round trip on V returns its coordinate space, with recovery supplied by c_V. For each f, the equation matrix(f)∘c_V=c_W∘f establishes the required compatibility. This compares the entire selected system of linear maps, including their compositions, rather than only matching vector values.

Now ask for lengths in Euclidean space. With basis b1=(1,0), b2=(0,2), coordinates (0,1) represent a vector of length 2; their ordinary coordinate length is 1. The linear comparison did not include the inner product. Carry it as a Gram matrix: here G=diag(1,4) and squared length is c^T*G*c. Retaining G lets us calculate the same vector’s length after changing coordinates.

A different question asks which linear maps preserve lengths. For a map from V to W represented by matrix M, with Gram matrices G_V and G_W, require M^T*G_W*M=G_V. This follows by comparing the squared length c^T*G_V*c of every input with (M*c)^T*G_W*(M*c) of its output. Use that condition to select the length-preserving maps. The earlier interpretation of arbitrary linear maps remains available when length preservation is not required.