3blue1brown's Linear algebra [closed]

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Hey I was going through 3blue1brown's video about linear transformation in space( at 4:00) and was wondering how the unit vectors get transformed such that the component in which they were 0 is now non-zero.

Eg) in the video vector [1,0] becomes [3,-2]

This matrix is adding a y component to the unit vector but how is it giving it a direction that our original never had?

I was curious how unit vector in x direction now gets a y component:

$$ \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} [1,0]'= $$ \begin{pmatrix} a + 0b \\ c + 0d \\ \end{pmatrix}$$= \begin{pmatrix} a \\ c \\ \end{pmatrix} $$$$ $$

This seems a bit strange to me as the x-component is added c times and made into the y-component.

side question: the matrix [[3,0],[-2,0]] transforms [0,0] to [0,0] and [0,1] or any [0,a] to [0,0] therefore points in this new space are not unique. This is not consistent with the stretching of the axis metaphor in the video right as this collapses a the line [0,a] to [0,0]?

$$ \begin{pmatrix} 3 & 0 \\ -2 & 0 \\ \end{pmatrix} [0,a]'=[0,0] $$

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1 Answer

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This matrix is adding a $y$ component to the unit vector but how is it giving it a direction that our original never had?

This seems to indicate that you expect vectors cannot change direction under a linear transformation. But this happens all the time: for example, rotations are linear transformations, and most rotations do change the directions of vectors. For example, in $\mathbb R^3$ or $\mathbb R^2$, you can take any two nonzero vectors and line them up to point in the same direction using a rotation.

Vectors that a linear transformation only scales, and doesn't change the direction of, are special: they're called eigenvectors for that transformation. Sometimes a transformation doesn't have any eigenvectors (for example, a 90 degree rotation in the plane.) I haven't watched the video link yet, but if it is discussing a transformation stretching certain vectors, it is almost certainly talking about an eigenvector (although it may be hidden in the context.)

therefore points in this new space are not unique.

The linear transformations that map the points uniquely are called nonsingular and again, that's a special property that not all linear transformations have.

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