Midterm 2

Question 1

Conditions of a subsapce

Answer 1

1. u + v ∈ V
2. ku ∈ V

Question 2

Are polynomials with real coefficients a vector space? (Pn)

Answer 2

Yes

Question 3

BAIS
For a vector space V, the vectors are called a basis of V if:

Answer 3

1. Linearly independent
2. Span V

Question 4

What is the dimension of a vector space?

Answer 4

elements in a basis

Question 5

Given a vector space V and a collection of vectors spanning V, if the vectors are linearly DEPENDENT, one of the vectors can be removed and ___

Answer 5

the set would still span V

Question 6

What do you know about any two bases of V?

Answer 6

they will always have the same # elements

dimension of a vector space is well defined

Question 7

The vectors v1…vn are a BASIS of V iff ___

Answer 7

every vector in V can be written in a unique way as a linear combination of v1…vn

Question 8

If a vector space has dimension n and the vectors u1, u2,…, un span V

Answer 8

Then they are a basis

Question 9

Uniqueness of a basis

Answer 9

If S = {v1,…vn} is a basis for a vector space V, then every vector v in V can be expressed in the form v = x1v1 + … + xnvn

Question 10

Coordinates of a vector space

Answer 10

Question 11

Change of coordinate

Answer 11

P and P-1 are the change of coordinate matrices

Question 12

Let V be an n dimensional vector space, and let {v1, v1, vn} be any basis.

Answer 12

f the set V has more than n vectors: it’s linearly dependent

If the set V has less than n vectors: it doesn’t span V

Question 13

Procedure for computing the change of coordinate matrix P B->C

Answer 13

1. Form matrix [C|B]
2. RREF
3. Resulting matrix [I|P B->C]

Question 14

Nullspace

Answer 14

set of vectors x that satify: Ax = 0

Question 15

Column space

Answer 15

span of the columns of A

(set of vectors b, such that Ax = b has a least 1 solution)

Question 16

Rowspace

Answer 16

span of the rows of A

Question 17

Row operations effect the null, row, or column space?

Answer 17

Change column space (not dimension)
Don’t change: nullspace & rowspace

Question 18

Do row operations change the span of the set of rows?

Answer 18

No

Question 19

Rank

Answer 19

common dimension of row space & column space

leading 1s in the general solution of Ax=0 (REF)

Question 20

rank(A) + dim(Null(A)) =

Answer 20

rank(A) + dim(Null(A)) = n

Question 21

Let A be an (mxn) matrix:
- If B is an invertible (nxn) matrix, then rank(BA) =
- If C is an invertible (mxn) matrix, then rank(BA) =

Answer 21

Let A be an (mxn) matrix:
- If B is an invertible (nxn) matrix, then rank(BA) = rank(A)
- If C is an invertible (mxn) matrix, then rank(BA) = rank(A)

Question 22

Determinant

Answer 22

scalar associated to a square matrix

Question 23

If rank(A) < n, det =
If rank(A) = n, det =

Answer 23

If rank(A) < n, det = 0
If rank(A) = n, det ≠ 0

Question 24

How to compute the determinant of a (2x2) matrix?

Answer 24

ad - bc

Question 25

How to compute the determinant of a (3x3) matrix?

Answer 25

Arrow technique

Question 26

Minor (Mij) of entry aij

Answer 26

the determinant of the submatrix that remains after the ith row and the jth column are deleted from A

Question 27

Cofactor of entry aij

Answer 27

Question 28

Cofactor expansion

Answer 28

obtained by multiplying the entries in any row/column by the cofactors and adding the resulting product = determinant

Question 29

Determinant properties: multiply all the elements of a single row/column by a fixed scalar (k)

Answer 29

det(B) = k det(A)

Question 30

Determinant properties: If two rows/columns are identical

Answer 30

det = 0

Question 31

Determinant properties: If a row/column is a sum of two vectors -> split it up into 2 seperate determinants

Answer 31

|a+b a a| = |a a a| + |b a a|

Question 32

Determinant properties: If we add to a row a multiple of another R1 + kR2

Answer 32

det stays same

Question 33

If two rows/columns of a matrix are linearly dependent

Answer 33

det = 0

Question 34

Relationship between determinants and transposes

Answer 34

det(A) = det(A^T)

Question 35

A matrix is invertible iff

Answer 35

det not equal 0

Question 36

Permutation

Answer 36

Reoredering of the elements

Question 37

det(AB) =

Answer 37

det(AB) = det(A)det(B)

Question 38

How can you computer the inverse of a matrix if you know its determinant?

Answer 38

A^-1 = (1/detA) * A

Question 39

Eigenvector

Answer 39

Ax = λx, x ≠ 0 Compute eigenvector: Null(A - λI)

Question 40

Eigenspace

Answer 40

collection of eigenvectors associated with each eigenvalue 1 ≤ dim eigenspace ≤ multiplicity λ in characteristic polynomials

Question 41

Characteristic polynomial

Answer 41

det(A - λI) λ: roots of the polynomial

Question 42

Diagonalizable matrix

Answer 42

if there exists an invertible matrix P and a diagonal matrix D such that: A = PDP-1

Question 43

The scalar λ is an eigenvalue of the (n x n) matrix A iff

Answer 43

det(A - λI) = 0

Question 44

Given a matrix A and the matrix obtained from A after a base change B = P-1AP

Answer 44

det(A - λI) = det(B - λI)

Question 45

The set of eigenvectors corresponding to a fixed eigenvalue a together with the zero vector form a subspace H of Rn

Question 46

Eigenvectors corresponding to different eigenvalues are __

Answer 46

linearly independent

Question 47

Let A be an nn matrix, then A is diagonalizable iff both conditions are satisfied:

Answer 47

Question 48

Complex number

Answer 48

a + bi

Question 49

Complex conjugate of a + bi

Answer 49

a - bi

Question 50

Modulus |z| of a+bi

Answer 50

(distance from point to origin)

Question 51

Complex number is real iff

Answer 51

it equals it's conjugate

Question 52

Properties of conjugates

Answer 52

conjugate of the sum is the sum of conjugates conjugate of the product is the product of conjugates

Question 53

If you know a polynomial/matrix has REAL entries and λ = a+bi

Answer 53

Then another eigenvalue must be λ = a - bi

Question 54

If A is an nn TRIANGULAR matrix, then the eigenvalues of A

Answer 54

are the entries on the main diagonal of A.

Question 55

B is similar to A (for square matrices)

Answer 55

if there is an invertible matrix P such that B = P-1AP

Question 56

Powers of diagonalizable matrices

Answer 56

A^K = P D^K P-1