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Tufts - Linear > Midterm 2 > Flashcards

Midterm 2 Flashcards

1
Q

Conditions of a subsapce

A
  1. u + v ∈ V
  2. ku ∈ V
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2
Q

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

A

Yes

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3
Q

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

A
  1. Linearly independent
  2. Span V
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4
Q

What is the dimension of a vector space?

A

elements in a basis

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5
Q

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 ___

A

the set would still span V

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6
Q

What do you know about any two bases of V?

A

they will always have the same # elements

dimension of a vector space is well defined

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7
Q

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

A

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

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8
Q

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

A

Then they are a basis

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9
Q

Uniqueness of a basis

A

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

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10
Q

Coordinates of a vector space

A
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11
Q

Change of coordinate

A

P and P-1 are the change of coordinate matrices

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12
Q

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

A

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

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13
Q

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

A
  1. Form matrix [C|B]
  2. RREF
  3. Resulting matrix [I|P B->C]
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14
Q

Nullspace

A

set of vectors x that satify: Ax = 0

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15
Q

Column space

A

span of the columns of A

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

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16
Q

Rowspace

A

span of the rows of A

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17
Q

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

A

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

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18
Q

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

A

No

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19
Q

Rank

A

common dimension of row space & column space

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

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20
Q

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

A

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

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21
Q

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) =

A

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)

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22
Q

Determinant

A

scalar associated to a square matrix

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23
Q

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

A

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

24
Q

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

A

ad - bc

25
Q

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

A

Arrow technique

26
Q

Minor (Mij) of entry aij

A

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

27
Q

Cofactor of entry aij

A
28
Q

Cofactor expansion

A

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

29
Q

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

A

det(B) = k det(A)

30
Q

Determinant properties:
If two rows/columns are identical

A

det = 0

31
Q

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

A

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

32
Q

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

A

det stays same

33
Q

If two rows/columns of a matrix are linearly dependent

A

det = 0

34
Q

Relationship between determinants and transposes

A

det(A) = det(A^T)

35
Q

A matrix is invertible iff

A

det not equal 0

36
Q

Permutation

A

Reoredering of the elements

37
Q

det(AB) =

A

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

38
Q

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

A

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

39
Q

Eigenvector

A

Ax = λx, x ≠ 0

Compute eigenvector: Null(A - λI)

40
Q

Eigenspace

A

collection of eigenvectors associated with each eigenvalue

1 ≤ dim eigenspace ≤ multiplicity λ in characteristic polynomials

41
Q

Characteristic polynomial

A

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

42
Q

Diagonalizable matrix

A

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

43
Q

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

A

det(A - λI) = 0

44
Q

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

A

det(A - λI) = det(B - λI)

45
Q

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

A
46
Q

Eigenvectors corresponding to different eigenvalues are __

A

linearly independent

47
Q

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

A
48
Q

Complex number

A

a + bi

49
Q

Complex conjugate of a + bi

A

a - bi

50
Q

Modulus |z| of a+bi

A

(distance from point to origin)

51
Q

Complex number is real iff

A

it equals it’s conjugate

52
Q

Properties of conjugates

A

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

53
Q

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

A

Then another eigenvalue must be λ = a - bi

54
Q

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

A

are the entries on the main diagonal of A.

55
Q

B is similar to A (for square matrices)

A

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

56
Q

Powers of diagonalizable matrices

A

A^K = P D^K P-1

Tufts - Linear (3 decks)

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