Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Linear Algebra 1554 > Midterm 2 > Flashcards

Midterm 2 Flashcards

1
Q

invertible matrix

A

a nxn matrix where AA-1 = I

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

inverse of matrix

A

A^-1 where AA^-1 = I

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

singular

A

not invertible

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

determinant of a 2x2 matrix

A

ad - bc

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

elementary matrix (E)

A

matrix performed by doing ONE elemtary row operation on an IDENTITY matrix
ALL elementary matrices are invertible because all row operations are reversible
– the inverse of an elementary matrix is another elementary matrix that’ll turn E back into I

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Row Equivalent Matrices

A

matrices that can turn into one another through a sequence of elementary row operations

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

nonsingular vs singular

A

invertible vs not invertible

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

inverse of a 2x2 matrix

A

(1/ad-bc)* [d -b ]
-c a

original: a b
c d

if determinant is 0, then A is NOT invertible because we can’t make an inverse!!

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Ax = b can be rewritten using inverses…

A

IF AND ONLY IF A is invertible
Ax = b
A^-1Ax = A^-1b
x = A^-1b

row reduction method is probably easier when it comes to finding the inverse of bigger matrices!

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

For all b in Rn, x = A^-1b is a unique solution

A

-invertible matrices have NO free variables
-has to be a unique solution

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

product of nxn invertible matrices ARE

A

INVERTIBLE

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

inverse of product is…

A

the product of the inverses in reverse order

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

elementary row operations performed on mxn matrix - the resulting matrix can be written as EA - what about multiple elementary row matrix?

A

Ek…E2E1A

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

method to find the inverse

A

row reduce A to the identity matrix while performing the same row operations on the identity matrix at the same time
○ [A | I] => [I | A-1]

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

matrix is invertible if and only if

A

ROW equivalent to the identity matrix == pivots in every row and column == onto and one-to-one (remember they are square matrices)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

linear transformations

A

mapping between two vector spaces (Rns) that preserves all vector addition and scalar properties

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

invertible linear transformations

A

T: Rn -> Rn (square matrix?)
if there is another linear transformation S: Rn -> Rn
WHERE
S(T(x)) = x for all x in Rn
T(S(x)) = x for all x in Rn
equivalent to saying that
A-1Ax = Ix

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Invertible Matrix Theorem (18) given A is a square nxn matrix, then the following statements are ALL equivalent

A

a) A is an invertible matrix
b) the columns of A form a linearly independent set
c) the columns of A span all of Rn
d) the transformation T: Rn -> Rn defined by T(x) = Ax OR the linear transformation x |-> Ax is one-to-one
e) the transformation T: Rn -> Rn defined by T(x) = Ax is onto
OR the linear transformation x |-> Ax maps Rn onto Rn
f) A has n pivot positions
g)A is row equivalent to the n x n identity matrix
h) Ax = 0 only has the trivial solution
i) the equation Ax = b has at least one solution for each b in Rn (doesn’t it also only have the unique solution for each b?)
j) There is an nxn matrix C such that CA = I
k) there is an nxn matrix D such that AD = I
l) A^T is an invertible matrix
m) columns of A form a basis of Rn
n) Col A = Rn
o) dimColA = n
p) rank A = n
q) Nul A = {0}
r) dim Nul A = 0
s) the number 0 is not an eigenvalue of A
t) the determinant of A is not 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Let A and B be square matrices: if AB = I, then…

A

A and B are both invertible
B = A^-1 & A = B^-1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

how many inverses can a matrix have

A

ONE - inverses of matrices are unique

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

How to determine if a linear transformation is invertible?

A

Let a matrix A represent the linear transformation
- if A is invertible, then the linear transformation is invertible!!
reflection through the y-axis is invertible but a projection is NOT

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

Adding 2 partitioned matrices A and B

A

A and B must be the same size, partitioned in the exact same way
add block by block

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

scaling partitioned matrices

A

scale block by block

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

multiplying 2 partitioned matrices A and B

A

column partition of A must equal row partition of B
OR the number of columns in partition A = number of rows in partition B
- multiply like regular matrices
so a 2x3 matrix times a 3x1 matrix will give you a 2x1 matrix

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

Inverses of Partitioned Matrices

A

A B ] [ X Y] = [In 0
0 C ] [Z W] 0 In]
AX + BZ = In
AY + BW = 0
CZ = 0
CW = In

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

Factorization of a matrix

A

expressing a matrix as the product of two or more matriches

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

row interchanges

A

swapping rows when row reducing

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

lower triangular matrix

A

entries above the main diagonal are all 0s

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
29
Q

upper triangular matrix

A

entries below the main diagonal are all 0s

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
30
Q

Algorithm for LU Factorization

A
  1. Reduce A to echelon form U by a sequence of row replacement operations, if possible
  2. Place entries in L such that the same sequence of row operations reduces L to I
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
31
Q

Why do we use LU Factorization

A

more efficient to solve a sequence of equations with the same coefficient matrix by LU factorization rather than row reducing the equations every single time

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
32
Q

Let A be an mxn matrix that can be row reduced to echelon form WITHOUT row exchanges !!!(interchanges?) - what is L and U?

A

L is a mxm lower triangular matrix with one’s on the main diagonal
U is mxn matrix that is the echelon form of A

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
33
Q

Rewriting Ax = b using A = LU

A

Ly = b
Ux = y
Ax = b -> L(Ux) = b

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
34
Q

How do we get U?

A

Row reduce A to echelon form using only row replacements that add a multiple of one row to another BELOW it

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
35
Q

Condition for LU Factorization

A

can be row reduced to echelon form without row exchanges??

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
36
Q

How do we get L?

A

take the row replacement operations you did on A when getting ecehlong form (find the elementary matrices that transform A into U)
and reverse the signs and input them in their respective spots on the mxm identity matrix
– replacing the 0s of identity matrix with the row replacement “coefficients”
– basically: after finding all the elementary row matrices, take their inverses
Ep…E1A = U
A = (Ep….E1)^-1U = LU
L = (Ep…E1)^-1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
37
Q

Using LU Decomposition

A
  1. Forward solve for y in Ly = b
    – modify row below using above rows
  2. Backwards solve for x in Ux = y
    – modify rows above using below rows
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
38
Q

Subset of Rn

A

any collection of vectors that are in Rn

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
39
Q

Subspace of Rn

A

A subset in Rn that has 3 properties:
- the zero vector is in H
- u +v in H (closed under addition)
■ cu in H (closed under scalar multiplication)
SUBSPACE can be written as the Span{} of some amount of linearly independent vectors (always linearly independent vectors? All combinations of linear vectors? can there be dependent vectors/)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
40
Q

Column Space of a Matrix A (mxn)

A

ColA
the subspace of Rm spanned by {a1…an}
essentially all the pivot columns!!

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
41
Q

Null Space of a Matrix A (mxn)

A

NullA
the subspace of Rn spanned by the set of all vectors x that solve Ax = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
42
Q

Basis for a Subspace H of Rn

A

A linearly independent set in H that spans H
– DOES NOT contain the zero vector (because it is linearly independent) unlike the span

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
43
Q

Standard Basis for Rn

A

{e1…en}

44
Q

if v1 and v2 are in Rn and H = Span {v1, v2}…

A

H is a subspace of Rn
v1 and v2 must be in Rn for this relation to work

45
Q

For v1 ….vp in Rn, the set of all linear combinations of v1…vp

A

subspace of Rn
Span{v1….vp} = subspace spanned by v1….vp

46
Q

is b in the column space of A?

A

same as asking:
is b a linear combination of A?
is B in the Span of A?

47
Q

is H a subspace of Rn? Or basis?

A

Asking if H has n linearly independent columns SO does H have no free variables?

48
Q

Subspaces vs Bases

A

Subspaces => Span{v1…vn}
– includes the 0 vector
Bases => {v1…vn}

49
Q

Defining a basis for column Space A

A

number of entries for each vector = number of rows in matrix A
number of vectors in the basis = number of pivot columns
what vectors can you include in the basis?
– scalar multiples??
– the identity matrix columns only if every column is pivotal in A

50
Q

Finding the Column Space

A

Row reduce the matrix
– row operations don’t depend on linear dependence relations!!
determine the pivot columns
create a basis/subspace using the pivot columns in the original matrix NOT THE ROW REDUCED ONE
– if every columns is linearly independent, then the elementary vectors are included in the column space, linear combinations of elementary vectors can get you any column of the original matrix!

51
Q

Find the Null Space

A

Determine all the free variables
rewrite the system in parametric vector form
vectors created in parametric vector form generate the null space

52
Q

Coordinates

A

weights that map our vectors to get to some point in the span of the vectors

53
Q

coordinate vector

A

suppose the set B = {b1….bp} is a basis for the subspace H. For each x in H, the coordinates of x relative to the basis B are the weights c1 … cp, such that x = c1b1 +… + cpbp and the vector in Rp

[x]_B = [ c1
.
.
.
cp ]
this is the coordinate vector of x (relative to B) or the B-coordinate vector of x

54
Q

Dimensions of a Subspace

A

dimH: the number of vectors in a basis of H
dim{0} = 0

55
Q

Rank of Matrix A

A

dimension of the column space of A
number of pivots in A

56
Q

Why we choose to write bases

A

Each vector in H can be written in only one way as a linear combination of the basis vectors

57
Q

A plane through 0 in R3 is

A

TWO-dimensional
3x3 matrix has 2 pivots

58
Q

A line through 0 in R2

A

ONE DIMENSIONAL
- is it important that the plane/line is through 0?
2x2 matrix A having one pivot

59
Q

Any two choices of bases of a non-zero subspace H have…

A

the same dimension

what is a bases??

60
Q

dim Rn
dim(Col A)
dim (Null A)

A

n
number of pivots
number of free variables

61
Q

dim(Col A)

A

rank A

62
Q

Rank Theorem

A

If A has n columns, then
rank A + dim(NulA) = n
number of pivots + number of free variables = number of columns

63
Q

Basis Theorem

A

-any two bases for a subspace have the same dimension (cardinality)
-many choices for the basis of a subspace

64
Q

Aij submatrix

A

delete the ith row and jith column of matrix A and the new submatrix are the remaining elements

65
Q

Determinant for a 2x2

A

|a b |
|c d |
ad - bc

c d |

66
Q

Cofactor expansion

A

find determinants of square matrices that are 3x3 and greater!

67
Q

Signs of cofactor expansion

A

depends on position of element aij in the matrix
Think checkerboard!! OR if i + j is even, then it’s positive!

68
Q

shortcut for finding determinants

A

row reducing to REF BUT be aware of effects of row operations on determinants!!
BECAUSE if you have a triangular matrix, you can just multiple all the numbers on the diagonal

69
Q

Columns Operations

A

same effect on determinants as row operations BECAUSE The determinant of A = determinant of A^T (transpose)

70
Q

Row operation effects on determinants

A

row replacement: nothing
row swap: multiple the determinant by negative one
row scale: multiply determinant by scale

71
Q

Summary of elementary matrices’ determinants

A

det EA = (detE) (detA)
det E is 1 if row replacement
-1 if row exchange
r if E is scaled by r
example: if a row is divided by k, then the determinant is multiplied by 1/k

72
Q

if A is invertible (terms of determinants)

A

det A != 0 because every column is pivotal!!
det A = (-1)^r times the products of the pivot in U when A is invertible

73
Q

if A is not invertible

A

det A = 0
at least one entry on the main diagonal of REF is 0

det A = 0 when A is not invertible
the rows are linearly dependent!! If A is a square, then the columns are ALSO linearly dependent!!

74
Q

Some properties of determinants

A

det A = det A^T
det AB = (detA) (detB)
det A^-1 = 1/ (detA)

75
Q

Parallelepiped

A

a parallelogram in Rn where n >2

76
Q

if A is a 2x2 matrix (area/volume)

A

area of the parallelogram determined by the columns of A is absolute value of determinant of A

77
Q

if A is a 3x3 matrix (area/volume)

A

area of the parallelepiped determined by the columns of A is absolute value of determinant of A

78
Q

row and column swaps and replacement DON’T

A

affect the absolute value of the determinant

79
Q

linear transformations on a parallelepiped/parallelogram

A

Area of T(S) = |det A| * {area/volume of S}
S: shape or figure
T: linear transformation determined by matrix A

80
Q

Probability vector

A

vector with NONNEGATIVE entries that sum to 1

81
Q

stochastic matrix

A

SQUARE matrix whose columns are probability vectors

82
Q

Markov chain

A

sequence of probability vectors together with a stochastic matrix {P} such that x1 = Px0, xk+1 = Pxk

83
Q

Steady State vector

A

A probability q such that Pq = q
EVERY stochastic matrix has a steady state vector

84
Q

regular stochastic matrix

A

stochastic matrix where some power of it will contain only STRICTLY positive entries
P^k where all entries >0

85
Q

How to find the next outcome of a Markov chain?

A

multiply P by xk to find xk+1

86
Q

How to find a steady state vector?

A

Pq = q
Pq - q =0
(P- I)q = 0
After finding a basis for the null space of (P- I)q = 0, remember to make sure that the column sum is 1
steady state vector IS a probability vector

87
Q

the initial state has what effect on the long term behavior of the Markov Chain

A

NO EFFECT

88
Q

Eigenvector of an nxn matrix A

A

nonzero vector x such that Ax = λx for some scalar λ

89
Q

eigenvalue of A

A

A scalar λ where there is a nontrivial solution x of Ax = λx

90
Q

eigenspace of an eigenvalue

A

contains the zero vector and all eigenvectors corresponding to λ
- any relaiton to A??

91
Q

Determine of a vector x is an eigenvector

A

1) NONZERO
2) Ax => see if the product is a scalar multiple of x

92
Q

Finding the eigenvector from an eigenvalue(7)

A

Solve (A - 7I)x = 0
do the parametric vector form of what you have left?? like solve for x??

93
Q

finding the eigenvalue λ

A

Solve (A -λI)x = 0 for a NONTRIVIAL solution
find the set of all solutions to the null space of (A - λI)

94
Q

Eigenvalues of a triangular matrix are….

A

the entries on the main diagonal!!

95
Q

0 is an eigenvalue of A if and only if A is …

A

NOT invertible!!
Ax = 0x
Ax = 0; x is a nontrivial solution if A is not invertible

96
Q

Eigenvectors that correspond to distinct eigenvalues are linearly independent

A

OPPOSITE IS NOT ALWAYS TRUE!!!
the identity matrix = eigenvectors are linearly independent but have the same eigenvalue

97
Q

Characteristic Polynomial

A

det(A - λI)

98
Q

Characterisitic Equation

A

det(A - λI) = 0

99
Q

Trace

A

sum of the diagonal entries in a matrix

100
Q

Algebraic Multiplicity of an Eigenvalue

A

number of times the eigenvalue shows up as roots of the characteristic polynomial

101
Q

Geometric Multiplicity of an Eigenvalue

A

dimension of Null (A - λI) for a given eigenvalue λ

102
Q

How to find eigenvalues?

A

Solve (A - λI) x = 0 for a nontrivial solution!
find the set of all solutions to the null space of (A - λI)

103
Q

finding the characteristic polynomial using trace and determinant for a characteristic polynomial of 2

A

λ^2 - λ(trace) + detA

104
Q

what do row operations do to eigenvalues
how can we determinant eigenvalues from its reduced forms?

A

THEY CHANGE THEM
WE CANT!!

105
Q

Properties of Invertible Matrices

A

(A^-1)^-1 = A
(AB)^-1 = B^-1A^-1
(A^T)^-1 = (A^-1)^T

106
Q

Transpose Equivalence for Determinants

A

If A is an nxn matrix, then det A^T = det A

107
Q

Eigenvectors for Distinct Eigenvalues

A

If v1….vr are eigenvectors that correspond to distinct eigenvalues λ1….λr of an nxn matrix A, the the set {v1…..vr} is linearly independent

Linear Algebra 1554 (3 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions