Midterm 2

Question 1

How many entries are in the vector representation of P^n

Answer 1

N+1, if you had P^4 for example it’d be a vector in R^5

Question 2

When doing vector space tests, how should u figure out if it’s a vector space

Answer 2

See what’s funny, does the set have some bounds? Is the vector addition or scalar multiplication special? Then, see how it doesn’t work based on the respective ones. 1-5 for vector addition, 1,4,6 for sets, 6-10 for scalar multiplication

Question 3

What are the axioms for the subspace test? (W is the subset, v is the subspace)

Answer 3

The zero vector of V is in W, for all u,v in W, u+v has to be in W, for all u in W, cu is also in W for any scalar c

Can’t an empty subset

Question 4

Is the 0 vector of R^n a subspace of R^n?

Answer 4

Yes

Question 5

Is the span of a subset of a vector space a subspace of the subset or a subspace of the vector space?

Answer 5

Subspace of the vector space

Question 6

What is the intersection of two subspaces that belong to the same vector space defined as

Answer 6

The set of all elements obviously within the vector space, but that are also within BOTH of the subspaces

Question 7

What is the union of two subspaces that belong to the same vector space defined as

Answer 7

The set of all elements obviously within the vector space, but that are also members of EITHER OR of the subspaces (could be both, technically)

Question 8

Is the intersection of W1 and W2 a subspace of V? Is the union?

Answer 8

The intersection is but the union isnt

Question 9

Nullspace definition

Answer 9

Set of vectors x in R^n such that Ax=0^m

Question 10

Column space definition

Answer 10

Set of vectors in R^m that are linear combinations of the columns of A. Aka the span of the columns/set of right hand side b such that Ax=b is consistent

Question 11

Dimension of nul(A) is…

Answer 11

free-variables=#non-pivot columns

Question 12

Nul(A) subspace of…

Answer 12

R^n

Question 13

Dimension of col(A)

Answer 13

pivot columns

Question 14

Col(A) subspace of…

Answer 14

R^m

Question 15

What points do nullA and rowA have in common? Domain?

Answer 15

R^n, only 0 vector

Question 16

Domain colA

Answer 16

R^m

Question 17

Do elementary row operations preserve the null space? What about the column space?

Answer 17

The null space is preserved but not the column space, same goes for the row space

Question 18

Basis for V (V is a vector space) definition

Answer 18

The subset of vectors have to be linearly independent and the vectors have to span the entire vector space

Question 19

Do you need to be invertible to be a basis? Is the empty subset a basis for V?

Answer 19

You do need it to be invertible

You won’t have a pivot on every column by definition if it’s an empty subset, so the empty subset is NOT a basis for V

Question 20

What is the dimension of a set that only includes the zero vector?

Answer 20

0

Question 21

For a basis, what should the number of vectors equal to?

Answer 21

The dimension, except for the empty set/0 dimension

Question 22

What is the standard basis for P^2?

Answer 22

{1,t,t^2}

Question 23

What is the standard basis for M2?

Answer 23

[1 0] [0 0] [0 1] [0 0]
{ [0 0], [1 0], [0 0], [0 1] }

Question 24

Let S={v1,v2,…,vp} be a set in V and let H =Span(S), if one of the vectors in the subset, vk, is a linear combo of ALL of the other vectors then..

Answer 24

H also equals Span{vk}, it’s still a subspace of the vector space that the subset belongs to

Question 25

Is rowA from the rref of A or the original matrix?

Answer 25

It is usually the rref or ref. It can only be the original matrix if NO rows were swapped

Question 26

NullityA =

Answer 26

DimNul(A) = #nonpivot columns

Question 27

Rank(A) =

Answer 27

DimCol(A) = dimRow(A) = #pivot columns

Question 28

What is the rank nullity theorem

Answer 28

rank(A)+nullity(A)=n=#of columns in A for a mxn matrix A

Question 29

For a given vector, V, how is it represented under a given basis? Vocabulary for the terms?

Answer 29

V=[V]_b*Pb [V]_b is the coordinates of v under the basis B. X1 on top then x2 down below and so forth, order matters Pb is the basis in matrix form

Question 30

[V]b=

Answer 30

Pb^-1v or the rref of [Pb|v] to [I|answer]

Question 31

How would you find coordinates of v under the standard basis if you know the change of basis coordinate?

Answer 31

Just decompose. Say vB is [3,2], it's just Pb*[v]b=[v]s=v

Question 32

Pb is

Answer 32

The change of coordinates matrix from b to the standard basis

Question 33

Pc<-B=

Answer 33

Pc^-1Pb=[[b1]c,[b2]c,...,[bn]c]

Question 34

If you know Pc<-B, what is [x]C using the coordinates of [x]b

Answer 34

[X]c=Pc<-B[x]B

Question 35

Is a square matrix singular if the determinant is 0 if it is not 2x2?

Answer 35

Yes. By definition, given a square matrix, the determinant being 0 means it's not invertible for any square dimension

Question 36

What is a minor?

Answer 36

If you deleted the ith/jth row and column, the determinant of that new matrix is the minor

Question 37

What is a cofactor?

Answer 37

The minor scaled by (-1)^(i+j)

Question 38

How do you find determinant using cofactors?

Answer 38

Go expand along a row or column and for each element, multiply it by the corresponding cofactor and add all the products up

Question 39

Whats an eigenvector?

Answer 39

It's a vector that preserves it's direction under the multiplication by A

Question 40

Whats an eigenvectors eigenvalue?

Answer 40

How much it scales for each multiplication by A