Chapter 6

Question 1

Orthogonal

Answer 1

inner product = 0

Question 2

Orthonormal

Answer 2

Orthogonal + norm = 1

Question 3

How to normalize a vector

Answer 3

u = 1 / ||v||

Question 4

Theorem: If S is an orthogonal set of nonzero vectors in an inner product space, then

Answer 4

S is linearly independent

Question 5

Conditions for being a basis for a vector space V

Answer 5

1. B spans V
2. B is linearly independent

Question 6

Coordinates relative to orthonormal bases

If S = {v₁, v₂…v_n} is an orthogonal basis for an inner product space V, and if u is any vector in V, then u =

Answer 6

Question 7

Coordinates relative to orthonormal bases

If S = {v₁, v₂…v_n} is an orthonormal basis for an inner product space V, and if u is any vector in V, then u =

Answer 7

Question 8

If W is a subspace of an inner product space V, then every vector (u) in W can be expressed as:

Answer 8

u = w₁ + w₂ = proj_Wu + proj_W⊥u

(perpendicular & parallel components)

Question 9

Every non-zero inner product space has an ___

Answer 9

orthonormal basis

Question 10

Why do you use the Gram-Schmidt process?

Answer 10

To convert a basis into an orthogonal basis

Question 11

Gram-Schmidt process procedure:

Answer 11

Question 12

If W is an inner product space, then:

1. Every orthogonal/orthonormal set of non-zero vectors can be ___

Answer 12

Every orthogonal/orthonormal set of non-zero vectors can be enlarged to an orthogonal/orthonormal basis for W

Question 13

Consistent linear system:

Answer 13

Consistent linear system: least squares error = 0

Question 14

Least squares solution of Ax = b

Answer 14

the vector x that minimizes ||b – Ax||

Question 15

How to find the least squares solution if A is not invertible?

Answer 15

If A not invertible ( least squares solutions) must parametrize → let t = constant in least squares error vector

Question 16

Least squares error vector

Answer 16

b - Ax

Question 17

Least squares error

Answer 17

||b - Ax||

Question 18

Normal equation

Answer 18

A^TAx = A^Tb

Question 19

If W is a subspace of inner product space V & b is a vector in V:

best approximation to b from W:

Answer 19

|| b – proj_Wb || < ||b - w||

Question 20

What is special about A^TA?

Answer 20

SYMMETRIC

Question 21

For every linear system Ax = b:

Answer 21

1. normal equation (A^TAx = A^Tb) is consistent
2. all solutions are least squares solutions

Question 22

If W in col(A) & x is any least squares solution of Ax = b, then the orthogonal projection of b on W is:

Answer 22

proj_Wb = Ax

Question 23

If A_{(m x n)} the following statements are equivalent:

Answer 23

1. The column vectors of A are linearly independent
2. A^TA is invertible

Question 24

If A^TA is invertible we can solve for least squares solution directly:

Answer 24

x = (A^TA)^-1 A^Tb

Question 25

If W in col(A), then orthogonal project of b on W is:

Answer 25

proj_Wb = Ax = A (A^TA)^-1 A^Tb

Question 26

What 4 properties must be satisfied in order to be an inner product?

Answer 26

1. Symmetry: =
2. Additivity: = +
3. Homogeneity: = k
4. Positivity: ≥ 0 and if = 0 then v = 0

Question 27

Norm

Answer 27

Question 28

Distance between two vectors

Answer 28

Question 29

Matrices: Inner product

Answer 29

Question 30

Standard inner product – Matrices

Answer 30

Question 31

Standard inner product – polynomials

Answer 31

dot coefficients

Question 32

Standard inner product – integrals

Answer 32

Question 33

Pythagorean theorem

Answer 33

If u orthogonal to v

Question 34

How to find angle between 2 vectors

Answer 34

Question 35

Least squares fit of a straight line: Ax = b

Answer 35

Question 36

Least squares fit of a QUADRATIC

Answer 36

Question 37

Mean square error

Answer 37

Question 38

Fourier series (general)

Answer 38

Question 39

Integration by parts

Answer 39