Thursday, April 19, 2018

Row, column, and null spaces for a matrix

Def

Example: make one in R3 with two successive orthogonal complement calculations

Theorem 4: An orthogonal set of non-zero vectors is LI.

Proof: Suppose lin comb is 0, take dot product with each vector in turn to show all coefficients 0.

Def: orthogonal basis

Theorem 5: Coordinates wrt orthog basis are easy to compute: Uc = y, multiply from left by *U*^{T}.

Def: orthonormal sets

Theorem 6: mxn matrix U has orthonormal columns means *U*^{TU} = *I*.

Theorem 7: Q has orthonormal columns (Q "orthogonal"): Orthogonal transformation preserves: length, dot product (in particular orthogonality)

Orthogonal decomposition theorem. W a subspace of Rn. Each y in Rn can be expressed as y = yhat + z where yhat is in W and z is in Wperp.

Specifally, if u1,...,up an orthogonal basis for W, yhat = ...

Example

projW y

Properties of orthog projections

1. projWy = y for y in W

2. Best approx theorem: y^-y closer to y than any other point v in W

Proof using Pythagoras

Example: find closest point in W (2 orthogonal vectors in R3) to y. (Student-supplied)

Theorem 10: Simpler formula for projW y in case of *orthonormal* basis
and matrix *UU*^{T} for projection.