# Tag Archives: matrix

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Once you’re comfortable with 2-arrays and 2-matrices, you can move up a dimension or two, to 4-arrays or 4-tensors. You can move up to a 3-array / 3-tensor just by imagining a matrix which “extends back into the blackboard”. Like … Continue reading

## Angle = Volume

This is trippy, and profound. The determinant — which tells you the change in size after a matrix transformation 𝓜 — is just an Instance of the Alternating Multilinear Map. (Alternating meaning it goes + − + − + − + … Continue reading

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## Jacobian

In the world of linear approximations of multiple parameters and multiple outputs, the Jacobian is a matrix that tells you: if I twist this knob, how does that part of the output change? Pretend that a through z are parameters, or knobs … Continue reading

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You know what’s surprising? Rotations are linear transformations. I guess lo conocí but no entendí. Like, I could write you the matrix formula for a rotation by θ degrees: But why is that linear? Lines are straight and circles bend. When you rotate something … Continue reading

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## Outer Product

Well I thought the outer product was more complicated than this. An inner product is constructed by multiplying vectors A and B like Aᵀ × B. (ᵀ is for turned.) In other words, timesing each a guy from A by his … Continue reading

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Bilinear maps and dual spaces Think of a function that takes two inputs and gives one output. The + operator is like that. 9+10=19 or, if you prefer to be computer-y about it, plus(9, 10) returns 19. So is the … Continue reading