Final

Question 0

det(Ei(r))

Answer 0

r

Question 1

det(Eij)

Answer 1

-1

Question 2

det(Eij(r))

Answer 2

1

Question 3

det(In)

Answer 3

1

Question 4

det(ABC…Z)

Answer 4

det(A)det(B)…det(Z)

Question 5

Determinant of a Triangular/Diagonal Matrix

Answer 5

product of its diagonal matrix

Question 6

Determinant: Type of matrices

Answer 6

Only square matrices

Question 7

rank(A) < n

Answer 7

det(A) = 0

no inverse

Question 8

Determinant of Transpose

Answer 8

= determinant of the original matrix

Question 10

Cofactor Expansion

Answer 10

Expand along a row or column

-1^(i+j)

Question 11

Property of Determinant

Answer 11

Can Commute

Question 12

Adjoint

Answer 12

Cofactor (without first term)
Transpose

A(adj(A)) = det(A)In
1/det(A)(adj(A)) = A^-1

Question 13

Powers of Matrices

Answer 13

A^k = QD^kQ^-1

Question 14

Eigenvalue on Triangular Matrix

Answer 14

Entries on diagonal are eigenvalues

det(A-bIn) = 0

Question 15

Rank Eigenvalues

Answer 15

For a matrix to have an eigenvalue, rank < n

Question 16

Type of Matrix for Eigenvalues

Answer 16

Square

Question 17

Characteristic Polynomial

Answer 17

det(A-xIn)

factor to find eigenvalues

Question 18

Eigenvectors

Answer 18

(A - yIn)X = 0

find basic solution for all eigenvalues (y)

Question 19

Diagonalizing

Answer 19

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

Question 20

If A has n distinct eigenvalues,…

Answer 20

Q is invertible

Question 21

Similar Matrices

Answer 21

A and diagonalized D

- same determinant, eigenvalues and characteristic polynomials

Question 22

det(A^-1)

Answer 22

1/det(A)

Question 23

Vector PQ

Answer 23

[q1-p1]

[q2-p2]

Question 24

Length of a vector

Answer 24

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

Question 25

Unit vector of v

Answer 25

v/|v|

Question 26

Dot Product

Answer 26

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

Question 27

Dot Product Properties

Answer 27

- distributive over addition - commutes with scalar multiples - dot product commutes

Question 28

Projection

Answer 28

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

Question 29

Equation of Line through two points

Answer 29

Point plus multiple of vector created by points

Question 30

Equation of a Plane

Answer 30

Normal vector multiply by change in coordinate

Question 31

cos(*)=

Answer 31

(u • v)/|u|•|v|

Question 32

|u x v| =

Answer 32

|u| • |v| sin(*)

Question 33

v • v =

Answer 33

|v|^2

Question 34

Equation of line

Answer 34

Point plus multiple of vector

Question 35

Cross Product (v x w)

Answer 35

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

Question 36

Properties of Cross Product

Answer 36

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

Question 37

Projection Transformation Matrix

Answer 37

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

Question 38

Rotation Transformation Matrix | counter-clockwise and clockwise

Answer 38

``` counterclockwise [cos* -sin*] [sin* cos*] clockwise [cos* sin*] [-sin* cos*] ```

Question 39

Definition Linear Transformation

Answer 39

T(v + w) = T(v) + T(w) | T(rv) = rT(v)

Question 40

Reflection Transformation

Answer 40

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

Question 41

Composition Transformations

Answer 41

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

Question 42

Inverse Transformation Conditions

Answer 42

if v =/ w then T(v) =/ T(w) | every vector must have a transformation

Question 43

Matrix for T^-1

Answer 43

A^-1

Question 44

Inverse Transformation Rotation

Answer 44

Other direction

Question 45

Inverse Transformation Projection

Answer 45

None

Question 46

Subset (U) Conditions

Answer 46

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

Question 47

Null(A)

Answer 47

set of all vectors in R^m such that Ax=0 | basic vectors to homogenous system

Question 48

Im(A)

Answer 48

set of all y vectors in R^n such that Ax=y | span is the set of the columns of the matrix

Question 49

Eigenspace

Answer 49

Null(A - yIn)

Question 50

Span

Answer 50

all linear combination of a set of vectors | any span is in R^n

Question 51

entries in vector

Answer 51

determines the R^n

Question 52

Linearly Dependent

Answer 52

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

Question 53

Linearly Independent

Answer 53

all parameters in the linear combination must be equal to 0 | det of the vectors =/ 0

Question 54

Basis

Answer 54

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

Question 55

Dimension

Answer 55

of elements in the basis

Question 56

Rank and Null(A)

Answer 56

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

Question 57

Rank and Col(A)/Row(A)

Answer 57

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

Question 58

Row Space

Answer 58

Span of the rows of A

Question 59

Basis of Row Space

Answer 59

Rows with leading ones in RREF

Question 60

Column Space

Answer 60

Span of columns of matrix | Same as Image

Question 61

Basis of Col(A)

Answer 61

Corresponding columns to the columns with leading 1s in RREF

Question 62

Determinant of an Inverse

Answer 62

1/det(A)

Question 63

det(AB) =

Answer 63

det(A)det(B)

Question 64

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

Answer 64

det(A) | - determinants commute since they are simply numbers

Question 65

A^-1 = | formula

Answer 65

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

Question 66

Cayley-Hamilton Theorem

Answer 66

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

Question 67

Conditions for T^-1

Answer 67

T(v) =/ T(w) | every vector must have a transformation

Question 68

Area of parallelogram

Answer 68

|v||w|sin(*)

Question 69

Reflection Matrix

Answer 69

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

Question 70

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

Answer 70

h^n det(A)