1) Ifu = 1...

v = 1...

are vectors in R then u v = 11 + ||u|| = u u dist(u v) = ||u v||

MTH 309 Y LECTURE 17.1 THU 2014.04.03

Recall:

2) u, v are orthogonal vectors if u v = 0.3) Pythagorean theorem: if u, v are orthogonal then

||u+ v||2 = ||u||2 + ||v||2

MTH 309 Y LECTURE 17.2 THU 2014.04.03

arbitrary basis = {v1 v}of V Rorthogonal basis = {w1 w}of V R

Gram-Schmidtprocessw1 := v1w2 := v2 w1v2w1w1w1w3 := v3 w1v3w1w1w1 w2v3w2w2w2 w := v w1vw1w1w1 w2vw2w2w2 w1vw1w1w1

5) If V R is a subspace and = {v1 v} is an orthogonalbasis of V then for w V we havew =

1...

where = w vv v .6) Gram-Schmidt process:

4) If V R is a subspace then an orthogonal basis of V isa basis of V that consists of vectors that are orthogonal toone another.

MTH 309 Y LECTURE 17.3 THU 2014.04.03

Motivation.

Problem. Find the flow rate of cars on each segment of streets.

Next: Orthogonal projections of vectors.

ADB

C120 cars

/hx1

x345 cars/hx5

x2

90 cars/hx4

72 cars/h

1) Recall: a matrix equation Ax = bhas a solution i b Col(A).2) In practical applications we may obtain a matrix equation Ax =b that have no solutions, i.e. where b Col(A).3) In such cases we may look for approximate solutions as follows:

replace b by a vector b such that b Col(A) anddist(b b) is as small as possible; then solve the equation Ax = b

MTH 309 Y LECTURE 17.4 THU 2014.04.03

Upshot.

bb

Problem. How to find the vector b?

Definition. Let V be a subspace of R. A vector w R isorhogonal to V ifw v = 0for all v V .

Vw

Proposition. If V = Span(v1 v) then a vector w is orthogo-nal to V i w is orthogonal to v1 v .

MTH 309 Y LECTURE 17.5 THU 2014.04.03

MTH 309 Y LECTURE 17.6 THU 2014.04.03

Definition. Let V be a subspace of R and let w R. Theorthogonal projection of w onto V is the vector projV w suchthat:1) projV w V2) the vector z = w projV w is orthogonal to V .

projV wV

zw

MTH 309 Y LECTURE 17.7 THU 2014.04.03

The Best Approximation Theorem.If V R is a subspace and w R, then projV w is the vectorof V which is the closest to w:dist(w projV w ) 6 dist(w v )for all v V .

If A is an matrix and b R then a least square solutionof the equation Ax = bis a solution of the equationAx = bwhere b = projCol(A) b.

MTH 309 Y LECTURE 17.8 THU 2014.04.03

Upshot.

Next. If V is a subspace of R and w R how can we computeprojV w?

projCol(A) b

b

MTH 309 Y LECTURE 17.9 THU 2014.04.03

Theorem. If {w1 w} is an orthogonal basis of a subspace Vof R then for u R we haveprojV u = u w1w1 w1

w1 + u w2w2 w2w2 + + u ww w

w

Corollary. If {w1 w} is an orthonormal basis of a subspaceV of R then for u R we haveprojV u = (u w1)w1 + (u w2)w2 + + (u w)w

MTH 309 Y LECTURE 17.10 THU 2014.04.03

Example.B =

1203

2452

4102 u =

1221

B is an orthogonal basis of some subspace V R4. Find projV u.

Example. Consider the following matrix A and vector u:A = 0 0 1 11 3 4 22 6 3 1

u = 030

Compute projV u where V = Col(A).

Note. In general, if V is a subspace of R and u R then to findprojV u we need to do the following:1) find a basis of V ;2) use Gram-Schmidt process to get an orthogonal basis of V ;3) use the orthogonal basis to compute projV u.

MTH 309 Y LECTURE 17.11 THU 2014.04.03

Example. Find least square solutions of the matrix equation Ax = bwhereA =

1 1 0 0 0 01 1 1 1 0 00 0 0 0 1 11 0 1 1 0 0

b =

907245120

MTH 309 Y LECTURE 17.12 THU 2014.04.03