12 Orthogonal Projections

Projections onto subspaces

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Then we can say that like this. We already have seen that vector x in R can be represented by a vector in V and a vector in complement orthogonal.

Visualizing a projection onto a plane

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The projection of x onto L, x - $proj_L x$ is orthogonal to L!!

Projection is closted vector in subspace

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Projection of x onto v is the closest vector in our subspace to x. It's closer than any other vector in v to our arbitrary vector x in $R^3$.

Least square approximation

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Let's say there's no solution to $Ax = b$.