This Stuff

Question 1

Scalars in subspaces

Answer 1

Scalars fuck with c*u in R unless stated or {u, v} is wide

Question 2

Subspace H

Answer 2

H = span{v1, … , vn}, for R^n
if zero vector

Question 3

Null A

Answer 3

All solutions for Ax=0

Question 4

Col A

Answer 4

ColA = span{ a1, a2, a3 }

Question 5

vector in NulA or ColA

Answer 5

NulA if Aw=0
ColA if lincomb of vector

Question 6

Bases

Answer 6

Linindept span

Question 7

Basis for NullA

Answer 7

REF
x = x1u + x2v
{u, v}

Question 8

Bases for ColA

Answer 8

Pivot columns in og

Question 9

Coordinate Vector

Answer 9

B * [x]B = x

Question 10

dimSubspace

Answer 10

# of columns in basis

Question 11

Change of Coordinates C <- B

Answer 11

[ b1 b2 | c1 c2 ]

Question 12

[x]C

Answer 12

P_(C<-B) * x = [x]C

Question 13

Using EiganValue

Answer 13

x + y = λx
x + y = λy

Question 14

Using EiganVectors

Answer 14

Ax = λx

Question 15

EiganSpace

Answer 15

Nul(A - λI)

Question 16

Characteristic Equation

Answer 16

det(A - λI) = 0
ad - bc = 0
x = EiganValue

Question 17

Similarity

Answer 17

A and B have the same Characteristic Equation

Question 18

Inner Product

Answer 18

u * v = uT v
IIuII IIvII cosθ

Question 19

Length of a vector

Answer 19

||v|| = sqrt(v^2)
||u - v|| = sqrt((u-v)^2)

Question 20

Unit Vector

Answer 20

if v / IIvII = 1
if v * v = 1

Question 21

2 Orthogonal Vectors

Answer 21

if u * v = 0
if IIu+vII^2 = IIuII^2 + IIvII^2

Question 22

Proj A onto B

Answer 22

ŷ = (B*A / A*A) * A

Question 23

y = ŷ + z

Answer 23

y = All vectors in IR
ŷ = All vectors in W
z = orth to W

Question 24

Diagonalization

Answer 24

A -> P D P^-1

Question 25

Orthogonal and Orthonormal

Answer 25

- Orthogonal = set orth from each other
- Orthonormal = unit vector set + orth from each other

Question 26

Gram Schmidt

Answer 26

v1 = x1
v2 = x2 - (x2v1 / v1v1) v1

Question 27

Orthonormal Basis

Answer 27

Run Gram Schmidt again
v1 = x1
v2 = x2 - (x2v1 / v1v1) v1

Question 28

Least Square Normal Equation

Answer 28

A^T Ax = A^T b

Question 29

Least Square Error

Answer 29

|| b - Ax̄ ||