Advanced Linear Algebra, Lecture 2.6: Matrices

Advanced Linear Algebra, Lecture 2.6: Matrices

If T is a linear map between finite-dimensional vector spaces, say from X to U, then we can represent T as a matrix once we pick an "input basis" for X and an "output" basis for U. The it's column are the coefficients of writing the image Tx_i as a linear combination of the output basis vectors. In other words, a_{ij}=(ℓ_j, Tx_i), the result of applying the j'th dual basis vector (in U') to the image Tx_i. This gives us a clean proof of why the matrix of the transpose map is simply the transpose of the matrix. We do several examples, such as the matrix of the projection onto the line y=x in R^2, and the matrix of the derivative map, both using several different choices of basis. We also see how we can always choose bases for X and U so that the matrix in block form has is the identity matrix in the upper-left, and zeros elsewhere (if there are any other entries).

[stevenvandokkum]

When encoding a linear map as a matrix, do we have to make sure that the linear map is surjective?
I was wondering this, as we want to express every element of the target space as the map of some elements in the domain.