3Blue1Brown's Linear Algebra Videos: Geogebra (Used for making plots): Relevant StackExchange Answer: Also relevant Quora Answer: Notes [1] By "gives a sense", I mean contains vectors that can form an orthogonal basis for the components of the vectors not in the ranges. [2] It isn’t really legal to plot R^2 and R^3 on the same set of axes like this, R^2 and R^3 are completely different spaces, but I’m showing them like this to provide some intuition and justification for all three parts of the SVD.