What parts of linear algebra are used in computer science?
The parts that you mentioned are basic concepts of linear algebra. You cannot understand the more advanced concepts (say, eigenvalues and eigenvectors) before first understanding the basic concepts. There are no shortcuts in mathematics. Without an intuitive understanding of the concepts of span and linear independence you won't get far in linear algebra.

Some algorithms only work with full rank matrices – Do you know what that means? Do you know what can make a matrix not full rank? How to handle this? You will have no clue if you don't know what linear independence is.

The Gaussian elimination algorithm that is used to solve linear equations can actually be numerically unstable if implemented improperly, and this is something that you might have to worry about in some cases. Without understanding the algorithm you won't know where the problem comes from and whether there's anything you can do about it – not at the level of algorithms for solving linear equations, but at the level of coming up with the correct linear equations to solve.

In short, don't be lazy, and take it on faith that these things are useful.