1. QR decomposition: A = QR Q is orthonormal R is upper triangular To find QR decomposition: 1.) Q: Use Gram-Schmidt to find orthonormal basis for column space of A 2.) Let R = Q T A

2. Find the QR decomposition of A = 14231423 1.) Use Gram-Schmidt to find orthonormal basis for column space of A

3. Find the QR decomposition of A = 14231423 1.) Use Gram-Schmidt to find orthonormal basis for column space of A col(A) = span, 1 4 2 3 { }

4. Find the QR decomposition of A = 14231423 1.) Use Gram-Schmidt to find orthogonal basis for column space of A col(A) = span, 1 4 2 3 { }

5. Find the QR decomposition of A = 14231423 1.) Use Gram-Schmidt to find orthogonal basis for column space of A col(A) = span, = span, 1 4 2 3 { } 1 2 1 2 { ?}?}

6. 4343 1 2 1 2

7. 4343 1 2 1 2 Find orthogonal projection of onto 4343 1 2 1 2

8. 4343 1 2 1 2 Find orthogonal projection of onto 4343 1 2 1 2 (1, 2)  (4, 3) (1, 2)  (1, 2) 1 2 1 2 4343 1 2 1 2 proj = 1(4) + 2(3) 4 + 6 1 2 + 2 2 1 + 4 = 1 2 1 2 1 2 1 2

9. 4343 1 2 1 2 Find orthogonal projection of onto 4343 1 2 1 2 (1, 2)  (4, 3) (1, 2)  (1, 2) 1 2 1 2 4343 1 2 1 2 proj = 1(4) + 2(3) 10 1 2 + 2 2 5 = 1 2 1 2 1 2 1 2

10. 4343 2424 1 2 1 2 Find orthogonal projection of onto 4343 1 2 1 2 (1, 2)  (4, 3) (1, 2)  (1, 2) 1 2 1 2 4343 1 2 1 2 2424 proj = 1(4) + 2(3) 1 2 + 2 2 = = 2 = 1 2 1 2 1 2 1 2

11. 4343 2424 1 2 1 2 Find orthogonal projection of onto 4343 1 2 1 2 (1, 2)  (4, 3) (1, 2)  (1, 2) 1 2 1 2 4343 1 2 1 2 2424 proj = =

12. 4343 2424 1 2 1 2 Find the component of orthogonal to 4343 1 2 1 2

13. 2 4343 2424 1 2 1 2 Find the component of orthogonal to 4343 1 2 1 2 − = 4343 2424 2

14. 4343 1 2 1 2 Short-cut for R 2 case:

15. Find the QR decomposition of A = 14231423 1.) Use Gram-Schmidt to find orthogonal basis for column space of A col(A) = span, = span, 1 4 2 3 { } 1 2 1 2 { } 2

16. 1 2 1 2 2 = √ 1 2 + 2 2 = √ 5 = √ (-1) 2 + 2 2 = √ 5 Find the length of each vector:

17. Divide each vector by its length: col(A) = span, = span, = span, 1 4 2 3 { } 1 2 1 2 { } 2 {}

18. {} col(A) = span, Q = A = QR

19. A = QR Q -1 A = Q -1 QR Q -1 A = R Q has orthonormal columns: Thus Q -1 = Q T Thus R = Q -1 A = Q T A

20. Find the QR decomposition of A = = QR R = Q -1 A = Q T A = = = 14231423 14231423