Brainscape's Knowledge GenomeTM

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


See full index

Linear Algebra > Linear Algebra > Flashcards

Linear Algebra Flashcards

1
Q

Solution

A

A list of values (s_1, s_2,…,s_n) that satisfy every equation of a linear system

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

Consistent

A

The solution set is nonempty

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

Inconsistent

A

The solution set is empty

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

Two linear systems are equivalent if

A

they share the same solution set

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

What are the options for the number of solutions a system can have?

A

1) Exactly one
2) Infinitely many
3) None

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

What does no solution look like in R2?

A

Two parallel lines

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

Two matrices are row equivalent if

A

There’s a sequence of elementary row operations that transform one matrix into another

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

In reduced row echelon for how many matrices are row equivalent?

A

Exactly one matrix

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

What is the rank of a matrix?

A

The number of pivot columns

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

What is the nullity of a matrix?

A

The number of free variable columns

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

If there are more variables than equations we…

A

Add a free variable

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

If there is a free variable then there are _______ solutions.

A

Infinitely many

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

Vectors u, v are equivalent if

A

u_1=v_1,…,u_n=v_n

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

Spanning set

A

The set of all linear combinations of a certain vector

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

Homogenous

A

The solution to the matrix equation Ax=b where b=0

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

Trivial solution

A

In the matrix equation Ax=b when x=0

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

The system Ax=0 has a nontrivial solution if and only if

A

it has at least one free variable

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

Linear independence

A

Each vector adds another dimension to the span

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

Linear dependence

A

When at least two vectors have the same span

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

Zero Matrix

A

An mxn matrix with all zero entries

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

Square Matrix

A

An nxn matrix

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

Main Diagonal

A

In a square matrix, it’s the values a_11, a_22,…,a_nn

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

Diagonal Matrix

A

Only the entries of the main diagonal are nonzero

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

Is the identity matrix a diagonal matrix?

A

Yes

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

Does AB=BA for any A,B

A

No

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

What sizes of two matrices must match for a product to be possible?

A

The column of the 1st matrix and the row of the 2nd matrix must match

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

What is the transpose of A?

A

A nxm(flipped) matrix where the 1st column=1st row and so on

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

Symmetric

A

A is an nxn matrix and A=A^T

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

Hermitian

A

A is an nxn matrix and A=A^*=A^(-T)

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

Quadratic form

A

A is a symmetric matrix and Q is a function defined by Q(x)=x^TAx

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
31
Q

Positive definite

A

x^tA, x>0 for nonzero x’s

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

Positive semidefinite

A

x^tAx, x>=0 for nonzero x’s

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

Negative definite

A

x^tAx, x<0 for nonzero x’s

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

Negative semidefinite

A

x^tAx, x<=0 for nonzero x’s

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

Indefinite

A

x^tAx, x>0 for all x

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

Nilpotent

A

k is a positive integer and A is an nxn matrix such that A^k=0(zero matrix)

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

Idempotent

A

An nxn matrix A where A^2=A

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
38
Q

Skew-symmetric

A

An nxn matrix A such that A^T=-A

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

Trace of A/Tr(A)

A

The sum of the main diagonals of A

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

A is invertible if

A

there exists an nxn matrix A^(-1) such that AA^(-1)=A^(-1)A=I_n

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

If A is an nxn matrix then

A

its inverse is unique

42
Q

A is orthogonal

A

A is an invertible nxn matrix and A^T=A^(-1) and therefore AA^T=A^TA=I_n

43
Q

Elementary matrix

A

A square matrix obtained by forming ONE elementary row operation on I_n

44
Q

If A is invertible then Ax=b must have how many solution(s)?

A

One(unique)

45
Q

Determinant of a 2x2 matrix A

A

a_11a_22-a_12a_21

46
Q

Submatrix of an nxn matrix A

A

Matrix obtained by removing the ith row and jth column

47
Q

(i,j)-cofactor of A

A

C_(ij)=(-1)^(i+j)det(A_(ij))

48
Q

How to find the determinant of matrices?

A

For 2x2 matrices, use the product of the first diagonal minus the product of the second diagonal. For larger square matrices, use the cofactor expansion across any row or column

49
Q

An nxn matrix A is invertible if and only if

A

det(A)!=0

50
Q

Cramer’s Rule

A

A is an invertible nxn matrix and b is a real number. Then there is a unique solution x in Ax=b given as x_i=(det(A_i(b)))/(det(A))

51
Q

Vector space

A

If ten axioms are true

52
Q

Vectors

A

Elements of a vector space

53
Q

Zero vector space

A

The set V that only contains 0

54
Q

Subspaces

A

Vector spaces that are formed from subsets of other vector spaces (A vector space V is a subset H of V)

55
Q

Properties of a subspace:

A

The zero vector is in H, it’s closed under addition (for u,v in H, we have (u+v) in H), and closed under scalar multiplication (for u in H we have cu in H)

56
Q

Zero subspace

A

The set H consisting of only the zero vector in V, H={0}

57
Q

H is a spanning set for V

A

Let v1,…,vk be in a vector space V, then H=span{v1,…,vk} is a subspace of V

58
Q

Null space of a mxn matrix A

A

Set of all solutions to the homogenous vector equation Ax=0: Nul(A)={x in Rn:Ax=0}
Set of all vectors x in Rn mapped to 0 in Rm by a linear transformation

59
Q

Column space of an mxn matrix A

A

The set of all linear combinations of the columns of A. Col(A)=span{a1,…,an} where A=[a1 … an]

60
Q

The column space of an mxn matrix A is a subspace of

A

Rm (rows)

61
Q

The null space of an mxn matrix A is a subspace of

A

Rn (columns)

62
Q

Linear transformation T:V -> W where V,W are vector spaces and

A

each x in V maps to a unique vector T(x) in W such that
1) T(x+v) = T(u) + T(v) for all u,v in V
2) T(cu) = cT(u) for all u in V

63
Q

Kernel (null space) of T

A

Let T:V->W be a linear transformation
The set of u in V such that T(u)=0 where 0 in W:
Ker(T)={u in V: T(u) = 0}

64
Q

Range of T

A

Let T:V->W be a linear transformation
The set of all b in W such that T(u)=b where u in V:
Rng(T={b in W:T(u)=b, u in V})

65
Q

Let S={v1,…,vk} be a set of vectors in a vector space V.
S={v1,…,vk} is a linearly independent set if

A

c1v1+…+ckvk=0 holds only when c1=…=ck=0

66
Q

Let S={v1,…,vk} be a set of vectors in a vector space V.
S={v1,…,vk} is a linearly dependent set if

A

c1v1+…+ckvk=0 holds for nonzero constants

67
Q

Let H be a subspace of a vector space V. An indexed set of vectors B={b1,…,bk} in V is a basis for H is

A

1) B is a linearly independent set
2) H=span{b1,…,bk} (B spans H)

68
Q

A basis for the Col(A) is formed from

A

the pivot columns of A

69
Q

Let B={b1,…,bn} be a basis for a vector space V and let x in V. The the B-coordinates of x are the constants c1,…,cn such that

A

x=c1b1+…+cnbn

70
Q

Let B={b1,…,bn} be a basis for a vector space V and let x in V. Assume c1,…,cn are the B-coordinates of x. Then the B-coordinate vector of x is

A

[x]_B =[c1
c2
.
.
.
cn]

71
Q

Let B={b1,…,bn} be a basis and [x]_B be the B-coordinate vector of x. For x=P_B[x]_B, the change of coordinates matrix from B to the standard basis in Rn is

A

P_B=[b1 …. bn]

72
Q

A vector space V is isomorphic to another vector space W if and only if

A

every vector space calculation in V can be accurately reproduced in W and vv

73
Q

Let B be a basis for a vector space V. The set {u1,…,un} in V is linearly independent if and only if

A

the set{[u1]_B,…,[un]_B}] is linearly independent in Rn

74
Q

Suppose B={b1,…,bn} is a basis for a vector space V. Then any set S in V with more than n vectors must be

A

linearly dependent

75
Q

If a vector space V has a basis with n vectors, then every other basis for V must have

A

n vectors

76
Q

Dimension of a vector space V

A

the number of vectors in the basis for V

77
Q

V is spanned by a finite set

A

V is finite-dimensional

78
Q

V is spanned by an infinite set

A

V is infinite-dimensional

79
Q

The dimension of the zero vector space is

A

0

80
Q

Let H be a subspace of a finite dimensional vector space V. Then

A

any linearly independent set in H can be expanded if necessary to a basis for H

81
Q

Let A be an mxn matrix. The row space of A is

A

the set of all linear combinations of the rows of A.
A=[r1
.
.
rn]
the Row(A)=span{r1,…,rn}

82
Q

If A and B are row equivalent, then

A

Row(A)=Row(B)

83
Q

If B is the row echelon form of A, then

A

the nonzero rows of B form a basis for Row(A) and Row(B)

84
Q

Let A be an mxn matrix. Then

A

dimCol(A)=dimRow(A)

85
Q

dimCol(A)=

A

=number of pivots of A

86
Q

dimRow(B)=

A

=number of nonzero rows of B

87
Q

dimNul(A)=

A

=number of free variable columns of A

88
Q

Let A be an mxn matrix. The rank of A is

A

the dimension of the column space of A:
Rank(A)=dimCol(A)=dimRow(A)

89
Q

Let A be an mxn matrix. The nullity of A is

A

the dimension of the null space of A:
Nullity(A)=dimNul(A)

90
Q

Let A be an mxn matrix, then Rank(A)=

A

=Rank(A^T)

91
Q

Rank-Nullity Theorem

A

Let A be an mxn matrix, then
Rank(A)+Nullity(A)=n

92
Q

If W is a subspace of Rn, then the orthogonal complement of W is

A

the set of all vectors in Rn that are orthogonal to every vector in W.
Wperp={v in Rn: v*w=0 for all w in W}

93
Q

Let W be a subspace of Rn, then

A

1) Wperp is a subspace of Rn
2) The only vector in common to W and Wperp is 0
3) The orthogonal complement of Wperp is W

94
Q

Let T:V->W be a linear transformation. The rank of T is

A

the dimension of the range of T:
Rank(T)=dimRng(T)

95
Q

Let T:V->W be a linear transformation. The nullity of T is

A

the dimension of the kernel of T:
Nullity(T)=dimKer(T)

96
Q

Rank-Nullity Theorem for Linear Transformations

A

Let T:V->W be a linear transformation. Then
Rank(T)+Nullity(T)=dimV

97
Q

A linear transformation T:V->W is one-to-one if

A

T maps distinct vectors in V to distinct vectors in W

98
Q

If Rng(T)=W, then T is called

A

onto

99
Q

A linear transformation T:V->W is one-to-one if and only if

A

Ker(T)={0}

100
Q

Let dimV=dimW=n. Then alinear transformation T:V->W is one-to-one if and only if

A

it is onto

101
Q
A

Linear Algebra (1 decks)

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