Brainscape's Knowledge GenomeTM

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


See full index

MATH 133 > Final > Flashcards

Final Flashcards

0
Q

det(Ei(r))

A

r

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

det(Eij)

A

-1

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

det(Eij(r))

A

1

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

det(In)

A

1

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

det(ABC…Z)

A

det(A)det(B)…det(Z)

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

Determinant of a Triangular/Diagonal Matrix

A

product of its diagonal matrix

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

Determinant: Type of matrices

A

Only square matrices

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

rank(A) < n

A

det(A) = 0

no inverse

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

Determinant of Transpose

A

= determinant of the original matrix

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

Cofactor Expansion

A

Expand along a row or column

-1^(i+j)

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

Property of Determinant

A

Can Commute

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

Adjoint

A

Cofactor (without first term)
Transpose

A(adj(A)) = det(A)In
1/det(A)(adj(A)) = A^-1
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

Powers of Matrices

A

A^k = QD^kQ^-1

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

Eigenvalue on Triangular Matrix

A

Entries on diagonal are eigenvalues

det(A-bIn) = 0

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

Rank Eigenvalues

A

For a matrix to have an eigenvalue, rank < n

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

Type of Matrix for Eigenvalues

A

Square

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

Characteristic Polynomial

A

det(A-xIn)

factor to find eigenvalues

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

Eigenvectors

A

(A - yIn)X = 0

find basic solution for all eigenvalues (y)

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

Diagonalizing

A

Q invertible
QD = AQ
Make eigenvectors of A the columns of Q
D = 0s with eigenvalue diagonal

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

If A has n distinct eigenvalues,…

A

Q is invertible

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

Similar Matrices

A

A and diagonalized D

- same determinant, eigenvalues and characteristic polynomials

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

det(A^-1)

A

1/det(A)

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

Vector PQ

A

[q1-p1]

[q2-p2]

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

Length of a vector

A

(x^2 + y^2)^(1/2)

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

Unit vector of v

A

v/|v|

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

Dot Product

A

Multiply terms that are in the same position and add them together.
vwcos(angle)

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

Dot Product Properties

A
  • distributive over addition
  • commutes with scalar multiples
  • dot product commutes
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

Projection

A

a parallel to w
b perpendicular to w
a = ((v•w)/|w|^2)w = projwV
b = v - a

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

Equation of Line through two points

A

Point plus multiple of vector created by points

30
Q

Equation of a Plane

A

Normal vector multiply by change in coordinate

31
Q

cos(*)=

A

(u • v)/|u|•|v|

32
Q

|u x v| =

A

|u| • |v| sin(*)

33
Q

v • v =

A

|v|^2

34
Q

Equation of line

A

Point plus multiple of vector

35
Q

Cross Product (v x w)

A

det of
[i j k]
[ v ]
[ w ]

36
Q

Properties of Cross Product

A

w x v = -(v x w)
u x (v + w) = u x v + u x w
v x (rw) = r(v x w)
v and w are orthogonal to v x w

37
Q

Projection Transformation Matrix

A

1/|w|^2 [a^2 ab]

[ab b^2]

38
Q

Rotation Transformation Matrix

counter-clockwise and clockwise

A 
counterclockwise
[cos*   -sin*]
[sin*     cos*]
clockwise
[cos*     sin*]
[-sin*     cos*]
39
Q

Definition Linear Transformation

A

T(v + w) = T(v) + T(w)

T(rv) = rT(v)

40
Q

Reflection Transformation

A

Projection transformation with w being equation of line and v being the perpendicular line
A = 2proj - v

41
Q

Composition Transformations

A

A: matrix for T
B: matrix for U
T(U(v)) = ABv

42
Q

Inverse Transformation Conditions

A

if v =/ w then T(v) =/ T(w)

every vector must have a transformation

43
Q

Matrix for T^-1

A

A^-1

44
Q

Inverse Transformation Rotation

A

Other direction

45
Q

Inverse Transformation Projection

A

None

46
Q

Subset (U) Conditions

A

0) 0 vector is in U
i) if u and v is in U then u + v is in U
ii) if u is in U then ru is in U

47
Q

Null(A)

A

set of all vectors in R^m such that Ax=0

basic vectors to homogenous system

48
Q

Im(A)

A

set of all y vectors in R^n such that Ax=y

span is the set of the columns of the matrix

49
Q

Eigenspace

A

Null(A - yIn)

50
Q

Span

A

all linear combination of a set of vectors

any span is in R^n

51
Q

entries in vector

A

determines the R^n

52
Q

Linearly Dependent

A

linear combination equal to zero with at least one parameter not equal to zero

53
Q

Linearly Independent

A

all parameters in the linear combination must be equal to 0

det of the vectors =/ 0

54
Q

Basis

A

linearly independent span
number of vectors =< n
number of vectors in a basis is the same as all bases

55
Q

Dimension

A

of elements in the basis

56
Q

Rank and Null(A)

A

m - r = # of vectors in basis of Null(A)

57
Q

Rank and Col(A)/Row(A)

A

rank(A) = dimension of Col(A)/Row(A)

58
Q

Row Space

A

Span of the rows of A

59
Q

Basis of Row Space

A

Rows with leading ones in RREF

60
Q

Column Space

A

Span of columns of matrix

Same as Image

61
Q

Basis of Col(A)

A

Corresponding columns to the columns with leading 1s in RREF

62
Q

Determinant of an Inverse

A

1/det(A)

63
Q

det(AB) =

A

det(A)det(B)

64
Q

det(B^-1)det(A)det(B)

A

det(A)

- determinants commute since they are simply numbers

65
Q

A^-1 =

formula

A

(1/det(A))(adj(A))

66
Q

Cayley-Hamilton Theorem

A

put matrix in for x and identity in for constants in the characteristic polynomial to get zero matrix

67
Q

Conditions for T^-1

A

T(v) =/ T(w)

every vector must have a transformation

68
Q

Area of parallelogram

A

|v||w|sin(*)

69
Q

Reflection Matrix

A

(1/a^2+b^2)[b^2-a^2 -2ab]

[ -2ab a^2-b^2]

70
Q

det(A^h) in a n x n matrix

A

h^n det(A)

MATH 133 (9 decks)

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