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Tufts - Linear > Final Exam > Flashcards

Final Exam Flashcards

1
Q

Do elementary row operations change the solution set?

A

No

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2
Q

When does a linear system have 0 solutions?

A

[0 0 0 1]
0x + 0y + … = 1

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3
Q

When does a linear system have 1 solution?

A

leading 1s = # variables

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4
Q

When does a linear system have infinite solution?

A

leading 1s < # variables
(paramterize)

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5
Q

Symmetric matrix

A
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6
Q

Span

A

set of all linear combinations

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7
Q

Linearly dependent

A

if there exists scalars (not all equal to 0) such that:
x1v1 + x2v2 + … + xnvn = 0

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8
Q

Linearly independent

A

if the only scalars are all 0 such that:
x1v1 + x2v2 + … + xnvn = 0

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9
Q

Homogeneous linear system

A

Ax = 0
Sends to zero vector

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10
Q

Non-homogeneous linear system

A

Ax = b
(translations of Ax = 0)

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11
Q

Homogenous system:
- If vector (v) is a solution to Ax = 0, then scalar (k) multiplied by v (kv) is what?

A

Also a solution to Ax = 0

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12
Q

Homogenous system:
- If vectors (v1, v2) are solutions to Ax = 0, then:
v1 + v2 is what?

A

v1 + v2 is also a solution to Ax = 0

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13
Q

Homogenous system:
- Any linear combination of homogenous solution (Ax=0) is what?

A

Also a solution

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14
Q

Systems:
- If vector (a) is a solution to Ax = 0
- If vector (c) is a solution to Ax = b
Then vector a+c is a solution to what?

A

homogenous solution + non-homogenous solution = non-homogenous solution

a+c is a solution to Ax=b

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15
Q

Systems:
- If vectors (a, c) are solutions to Ax = b
Then vector a-c is a solution to what?

A

Vector a–c is a solution to Ax=0 (homogenous)

non-homo – non-homo = homo

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16
Q

Linear transformation

A
  1. T(v+w) = T(v)+T(w)
  2. T(kv) = kT(v)
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17
Q

Invertible matrix

A

det ≠ 0

  • Equation Ax=b has a unique solution
  • Matrix has full rank
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18
Q

Can non-square matrices have inverses?

A

Yes, but only on one side

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19
Q

Subspace

A
  1. u+v in V
  2. ku in V
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20
Q

Basis

A
  1. Linearly independent
  2. Span
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21
Q

How can you determine the dimension of a vector space?

A

elements in basis

UNIQUE

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22
Q

If a vector space has dimension n, then n linearly independent vectors must be a ___

A

basis

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23
Q

If a vector space V has dimension n, and collection of vectors span(V) then ___

A

vectors are a basis

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24
Q

Coordinates of a vector space
- v = arbitrary vector in V
- Basis: {v1,v2…vn}

A

a1v1 + a2v2 + … + anvn = v

Collection of scalars [a1 a2 … an]

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25
Q

Find the matrix that represents the transformation (T) from basis A to basis B

A
  1. Compute image of each basis vector of A:
    T(a1), T(a2) … T(an)
  2. Re-write in terms of basis B:
    T(a1) = x1b1 + … + xnbn
  3. From matrix using coefficients (x1…xn) as columns
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26
Q

Characteristic polynomial

A

det(A – λI)

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27
Q

How to find eigenvectors given λ?

A

null(A – λI)

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28
Q

Fundamental formula relating rank & dim(Null)

A

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

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29
Q

Rank

A

linearly independent rows of the matrix
# leading 1s in RREF

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30
Q

If a matrix is (mxn) what is it’s rank?

A

Whichever is smallest m or n

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31
Q

Nullspace

A

Solution to Ax=0

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32
Q

Column space

A

span of the columns of A

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33
Q

How to determine the column space of a matrix?

A
  • RREF A → find the columns with leading 1s
  • Col(A) = span(columns with leading 1s in ORIGINAL matrix A)
  • Basis for Col(A) = {columns of original matrix A which correspond to leading 1s in RREF A}
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34
Q

Row space

A

span of the rows of A

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35
Q

How to determine the row space of a matrix?

A
  • RREF A
  • Row(A) = span (RREF’s rows of A)
36
Q

Do row operations preserve the row space?

A

Yes!

37
Q

Do row operations preserve the column space?

A

No!

But the dim(col(A)) yes

38
Q

Determinant of a triangular matrix

A

product on main diagonal

39
Q

If rank < n, then determinant = ?

A

det = 0

40
Q

What is the minor? (Mij)

A

determinant submatrix remains after the ith row & jth column are deleted

41
Q

What is the cofactor? (Cij)

A
42
Q

Determinant properties:
- Multiply all elements of 1 row/column by k

A

k * det

43
Q

Determinant properties:
If a row/column is a sum of two vectors

A

det can be written as the sum of the 2 determinants with each of the 2 row vectors and the remaining rows/columns as in the original

44
Q

Determinant properties:
If 2 rows/column are identical

A

det = 0

45
Q

Determinant properties:
Add to a row/column a multiple of another row

A

det stays SAME

46
Q

Determinant properties:
If 2 rows/column are linearly dependent

A

det = 0

47
Q

det(AB) =

A

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

48
Q

Eigenvector (v)

A

Av = λv
v ≠ 0

49
Q

Does changing the basis of a linear transformation change its characteristic polynomial?

A

No

50
Q

The set of eigenvectors corresponding to a single eigenvalue together with the zero vector form a __

A

subspace of Rn

51
Q

Eigenvectors corresponding to different eigenvalues are what?

A

Linearly independent

52
Q

Diagonalisable if

A
  1. dim(eigenspace) = multiplicity of eigenvalue for each λ
  2. able to perform a change of coordinate that brings matrix to diagonal form D = P^-1 AP
53
Q

Every eigenvalue must have at least how many eigenevctors?

A

1

54
Q

If a charactertistic polynomial has n DIFFERENT eigenvalues (mult = 1) then?

A

diagonalizable

55
Q

Complex #

A

z = a + bi

56
Q

i^2 =

A

-1

57
Q

Complex conjugate

A

z(bar) = a - bi

58
Q

Moduluz: |z|

A

distance from point to the origin

59
Q

Given an (nxn) matrix with REAL entries, if a+bi is an eigenvalue with eigenvector (v) then what is another eigenvalue/eigenvetcor?

A
60
Q

Orthogonal

A

dot product = 0

61
Q

Orthonormal

A

dot product = 0
length = 1

62
Q

How to normalize a vector?

A
63
Q

How to compute the length of a vector? ||v||

A
64
Q

Orthogonal complement

A

set of vectors that are orthogonal to every vector in subspace H

65
Q

A vector can be written as the sum of its orthogonal projection and what?

A
66
Q

Orthogonal projection

A

v hat

67
Q

Gram-Schmidt Orthogonalisation Proces

A

transform a basis → orthogonal basis

68
Q

What does a least squares approximation do?

A

Solves the equation Ax=b as closely as possible

least squares = vector y such that |b-Ay| is as small as possible…

69
Q

2 methods for computing least squares approx

A
  1. Find the projection b(hat) of b on the space spanned by the columns of A, then solve the equation Ax=b(hat).
  2. Solve A^T Ax= A^T b
70
Q

Line of best fit to n data points

A

Find the values of a,b in the linear equation
y = a + bx that minimize the distance between the n data points

71
Q

Orthonormal matrix

A
72
Q

An orthonormal change of coordinate transforms a symmtetric matrix into what?

A

Into another symmetric matrix

73
Q

A square symmetric matrix with real entries has __ eigenvalues

A

real eigenvalues

74
Q

If v1 and v2 eigenvectors of A with different eigenvalues, then

A

v1 and v2 are orthogonal

75
Q

An nxn symmetric matrix with real entries is always diagonalizable over the complex numbers in an ____ basis.

A

orthonormal basis

76
Q

Given a symmetric matrix (A=A^T), we find an orthonormal matrix P and a diagonal matrix D both with real entries such that:

A
77
Q

What is thefFirst step to diagonalization of symmetric matrices?

A

Show that all eigenvalues are real

78
Q

Positive definite

A

symmetric matrix whose eigenvalues are all positive

79
Q

Are all positive definite matrices invertible?

A

Yes
det = +

80
Q

Given any invertible matrix, ___ is positive definite

A

Cholesky decomposition

81
Q

For any nxn symmetric matrix A, the following conditions are equivalent (and therefore any imply that the matrix is positive definite):

A
82
Q

Cholesky Decomposition

A
83
Q

Matrix representation of a quadratic function

A

f(x) = x^T A x

84
Q

Given a quadratic function with f(x) = x^T A x where A is symmtric. The origin is a local min if?

A

A is positive definite

85
Q

Given a quadratic function with f(x) = x^T A x where A is symmtric. The origin is a local max if?

A

–A is positive definite

86
Q

Given a quadratic function f(x) = x^T A x and a circle ||x||=1, what is the max/min value of f?

A

Max = largest eigenvalue of A
Min = smallest eigenvalue of A

Tufts - Linear (3 decks)

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