← MATH 01210 · Linear Algebra
MATH 01210 · Linear maps & eigenstuff
Orthogonality / projections (often)
Definition
Vectors u⊥v if u·v=0. Orthogonal sets are independent (if nonzero). Projection of b onto a: proj_a b = ((b·a)/(a·a)) a. Least squares solves AᵀAx=Aᵀb.
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Orthonormal bases make coordinates easy (Fourier-style coefficients).
QR and Gram–Schmidt build orthonormal bases from independent sets.
Orthogonal matrices: QᵀQ=I, preserve lengths/angles.
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Attempt each problem first, then open the worked solution.
1.Project (1,1) onto span{(1,0)}.