Week 10

Question 1

Column Space?

Answer 1

-Linear span of columns of M , before RREF

subspace of R^m (where m is rows)

Question 2

Row Space?

Answer 2

-Lin span of transpose rows of M before RREF

subspace of R^n (where n is columns)

Question 3

Null Space?

Answer 3

the set of solutions of the system Mx=0

subspace of R^m (where m is rows)

Question 4

What are the basis for CS and RS?

Answer 4

-CS = Pre RREF, whatever are the columns making up it
-RS =Post RREF

Question 5

Basis for null space?

Answer 5

the same

Question 6

How to show linear makeup from basis?

Answer 6

-RREF basis columns and column that is made up of one and the equation we get e.g. c1 -c2 +c3 = 0 , we can show it is

Question 7

Rank Nullity?

Answer 7

dim(RS(M) + dim (CS(M) = dim (R^n) where n are columns

Question 8

How to prove RS(A) and NS(A) are orthogonal complements?

Answer 8

• rank nullity theorem
  -dot product of any vector of each = 0

Question 9

Null Space Cartesian Equation?

Answer 9

-use basis for RS(A) as orthogonal

and plug into eq

Question 10

What is null space?

Answer 10

all x that solves Ax = 0

Question 11

How to get cartesian description of null space?

Answer 11

-we use basis of row space transposed and multiply by vector (x1,x2,x3…) = 0

Question 12

How to get cartesian description of Row Space?

Answer 12

-we use basis of null space transposed and multiply by vector (x1,x2,x3…) =0

Question 13

How to get cartesian description of column space?

Answer 13

-Transpose matrix then RREF, then paramteric equation vectors multiplied by x = 0 then we get our cartesian equations

or if we have 2 vectors in basis, we just use the normal from that by cross product

Question 14

How to get the cartesian equation of subspace that spans the set X

Answer 14

-same as cartesian description of column space

Question 15

How to show a system is consisyent using column space?

Answer 15

Column space condition: b∈CS(A), meaning
b can be written as a linear combination of the columns of A.

Question 16

condition for a set of vectors to provide a basis in a R^n?

Answer 16

det ≠ 0

Question 17

Coordinates (V)c with respect to basis c

Answer 17

you just use vectors of c and solve a(v1) + b(v2) = (v)

Question 18

How to show a set of functions is linearly independent?

Answer 18

c1f1(x) +c2f2(x) = z(x)

only if c1 and c2=0

Question 19

How to show a set spans a subspace for functions?

Answer 19

That the subspace can be written as combination of the functions

Question 20

bilinearity?

Answer 20

⟨u,βb+γc⟩=β⟨u,b⟩+γ⟨u,c⟩

use example and just plug in inner product <u,v> to get end

linearity on right is keep vector on left and vice versa for on right

Question 21

Symmetry?

Answer 21

<v,u> = <u,v>

Question 22

Positivity?

Answer 22

<u,u> ≥0 or if <u,u> = 0 only if u=0

Question 23

<n,n>?

Answer 23

ll n ll ^2