Question

The examples form the book are: zero map, identity linear map, differentiation linear map, integration linear map, multiplication by x² linear map, backward shift linear map, from R³ to R², or generally  from Fⁿ to F^m linear map.

Can you give an example (not one of the examples already in the book) of a linear map which is not injective?

Answer 1

1. If A is a real m × n matrix, then A defines a linear map from Rⁿ to R^m by sending the column vector x ∈ Rⁿ to the column vector Ax ∈ R^m. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner.

2. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f : V → W to dim_F(W) × dim_F(V) matrices in the way described in the sequel are themselves linear maps.

3. The expected value of a random variable (which is in fact a function, and as such a member of a vector space) is linear, as for random variables X and Y we have E[X + Y] = E[X] + E[Y] and E[aX] = aE[X], but the variance of a random variable is not linear.