if a matrix is a transformation, then how data can be interpretated in terms of linear mapping?

Question

We can interpret a matrix as a linear mapping in linear algebra using matrix representation. However, in machine learning or deep learning, we represent input data as a matrix. Then, how can this data be interpreted in terms of linear mapping?

Answer 1

If $X$ is $n \times p$ where $n$ is the number of samples and $p$ is the number of features (a.k.a. covariates or predictors), then each column of $X$ is a vector in $\mathbb{R}^n$ that represents your data's information about a particular feature. Then you could interpret $X$ as mapping the first standard basis vector $(1,0,\ldots,0)$ to the first feature vector, the second standard basis vector $(0,1,0,\ldots,0)$ to the second feature vector and so on.

This interpretation of $X$ as a mapping is not particularly useful, but thinking about the column space as a subspace of $\mathbb{R}^n$ is useful when thinking about linear regression, PCA, etc.