Linear Algebra Mastery Hub: The Industry Foundation Practice

In the context of data science, what is the primary advantage of representing a dataset as a matrix and performing operations like singular value decomposition (SVD)?

Consider a scenario in machine learning where you are training a linear regression model. The design matrix $X$ has dimensions $m \times n$, where $m$ is the number of samples and $n$ is the number of features. If $m < n$ (more features than samples), what is the most likely consequence for the standard least squares solution $\hat{\beta} = (X^T X)^{-1} X^T y$?

In the context of neural networks, what does the concept of "gradient vanishing" primarily relate to, and how can linear algebra concepts help in understanding and mitigating it?