1. The first 2 steps of the Gram Schmitt Process
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2. It is often helpful to take a basis of a subspace and write it in a form such that the vectors are orthonormal
To do this just subtract off the component of the vector that is parallel to the previous vectors in the basis. Repeat this process for each vector in the basis.

3. A Geometric view

4. Example 1a,b

5. 1a

6. 1b Solution

7. The Gram-Schmidt Process

8. Problem 6
Perform the Gram-Schmitt process

9. Problem 6 Solution

10. Problem 4

11. Solution to problem 4

12. QR factorization List the columns of the orthornomal matrix found in Gram-Schmidt Then use the formula below to find a matrix R that when multiplied by Q generates M (the original matrix) This factoring has some applications in higher math.
Note: This formula works
in 2 D, R can be seen as a
series of coordinate
Vectors we will use this for
higher dimensions

13. Problem 16
Find the QR factorization of the following matrix

14. Solution to 16 M=
Note: the vectors are already orthogonal so only divide by the length.
Create R either by formula or by coordinate vectors.

15. Problem 20
Find the QR factorization of

16. Problem 20 solution M=
One shortcut. If rref of M is I then I is an orthonormal basis of M and Gram-Schmitt is not required

17. Problem 21
Find the QR factorization

18. Problem 21 solution M=
M = QR therefore Q-1M = R

19. Homework p odd