I can’t recommend the video series Essence of Linear Algebra highly enough. These Khan Academy videos are a solid review if you haven’t studied linear algebra in a while. (I’m not sure whether they’re a good first introduction to the topic, but they’re certainly better than learning linear algebra as a series of operations on matrices that just happen to have strange magical powers.)

When I first learned linear algebra, it was as a system of linear equations. That viewpoint explained how the math might be applied, but not why. These videos develop a geometric intuition for linear algebra as describing linear transformations of space.

For example, I always had trouble understanding why a determinant is so important. It’s used in many places, but what was so special about multiplying diagonally and then subtracting? It was just a rule to memorize.

The video on determinants develops the intuition that the determinant measures how much a linear transformation stretches or squishes space. If the transformation shrinks space to a smaller dimension, the determinant is 0. If you shrink two-dimensional space to one dimension, the area after the transformation — a line — is 0. There you go.

I know this reads like a gushing Goodreads review, but for real: After watching these videos, I can visualize how linear algebra works on a geometric level, and that has given me a deeper understanding of why it is so important in statistics and engineering.

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