Section 21

Orthogonality and projections

intermedioviz

u\perp v\iff u\cdot v=0

Characterizes perpendicular vectors.

intermedioviz

\operatorname{proj}_b(a)=\frac{a\cdot b}{\|b\|^2}b

Projects a vector onto the direction of another.

intermedioviz

Q^TQ=I\Rightarrow Q^{-1}=Q^T

Preserves lengths and angles.

avanzadoviz

u_k=v_k-\sum_{j<k}\operatorname{proj}_{u_j}(v_k)

Builds an orthogonal or orthonormal basis from independent vectors.

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Section 21

intermedioviz

u\perp v\iff u\cdot v=0

Characterizes perpendicular vectors.

intermedioviz

\operatorname{proj}_b(a)=\frac{a\cdot b}{\|b\|^2}b

Projects a vector onto the direction of another.

intermedioviz

Q^TQ=I\Rightarrow Q^{-1}=Q^T

Preserves lengths and angles.

avanzadoviz

u_k=v_k-\sum_{j<k}\operatorname{proj}_{u_j}(v_k)

Builds an orthogonal or orthonormal basis from independent vectors.