Math 19B - Final Exam Review

Front 1

If a transformation is through an n x m matrix, what is the dimensional change?

Back 1

From R^m to R^n

Front 2

When is k |x|
|x|
a dilation and when is it a contraction?

Back 2

Dilation - when K is greater than 1
Contraction - when K is between 0 and 1

Front 3

How can you describe a projection in words?

Back 3

proj(x) means turning x into something x(parallel) and x(perpendicular)

2.1

Front 4

How do you calculate a projection onto a line spanned by a vector?

Back 4

1. Normalize the vector
2. proj(x) = (x•u)u

Front 5

How do you describe ref(x) algebraically?

Back 5

ref(x) = 2proj(x) - x = 2(x•u)u - x

Front 6

How do you describe a reflection in words?

Back 6

Flipping a vector over something = x(parallel) - x(perpendicular)

Front 7

What conditions have to be fulfilled in order for a matrix to be invertibel?

Back 7

1. It must be square
2. rref(A) = 1

Front 8

If T is transofmred from R^m to R^n, describe the image and the kernel.

Back 8

im(T) is the subset of the codomain R^n
ker(T) is the domain of R^m

Front 9

How do you describe the inverse of a 2x2 matrix, A = |a b|
|c d|

Back 9

A^-1 = 1/(det(A)) |d -b|
|-c a|

Front 10

If it is in the form y1 = ax1 + bx2 + cx3, what form is the inverse in?

Back 10

Inverse is in the form, x1 = ay1 + by2 + cx3

Front 11

What is hte kernel of an invertible nxn matrix?

Back 11

ker = {0}

Front 12

Is im(A) necessarily equal to im(rref(A))?

Back 12

No

Front 13

For an invertible nxn matrix, is im(A) = R^n?

Back 13

im(A) = R^n

Front 14

What is the rank?

Back 14

Dimension of the image.

Front 15

Is the image dependent on kernel?

Back 15

No

Front 16

What does it mean to be orthogonal?

Back 16

Dot product is 0

Front 17

What does orthonormal mean?

Back 17

Unit vector are orthogonal to one another.

Front 18

Are orthonormal vectors linearly independent?

Back 18

Yes

Front 19

What do orthnormal vectors form?

Back 19

Basis of R^n.

Front 20

How would you describe proj(x) in words?

Back 20

An orthogonal projection amounts to porjectings it onto the x-axis, then onto the y-axis, then adding the resultant vectors.

Front 21

How does you describe proj(x) algebraically?

Back 21

(u1 • x1) u1 + ... + (uN • xN) uN

Front 22

What is orthogonal complement?

Back 22

The set of vectors in R^n that are orthogonal to all vectors in V.

Front 23

What is the orthogonal complement the kernel of?

Back 23

The orthogonal projection on to V.

Front 24

What is the dimension of the subspace + the dimsneion of the kernel equal to?

For example, what is the kernel of a plane in R^3?
What is the kernel of a line in R^3?

Back 24

0

Line: 1 + 2 = 3
Plane: 2 + 1 = 3


(4.1)

Front 25

What is V ∩ V⊥?

Back 25

0

Front 26

What condition do 2 vectors x and y fit ≡ they are orthogonal complements?

Back 26

||x + y||^2 = ||x||^2 + ||y||^2

Front 27

Which is always greater than or equal to the other: x or proj(x)?

Back 27

x by the Pythagorean Theorem

Front 28

What is the angel between 2 vectors?

Back 28

arc cos (x•y/(||x|| ||y||))

Front 29

What is the correlation coefficient between two vectors?

Back 29

r = cos(Θ) = (x•y/(||x|| ||y||)

(4.2)

Front 30

What is another way to write v1 • v2 using Θ?

Back 30

v1•v2 = ||v1|| ||v2|| cos(Θ)

Front 31

If basis of V is v1 and v2, then how would you find the orthonormal basis?

Back 31

1. u1 = normalized v1
2. u2 = u2 - (u1•v2)u1

(v3 would equal u3 - (u1•v3)u1 - (u2•v3)u2/ ||"|| )

Front 32

Describe the relationship between basis and image with the kernel.

Back 32

Basis is found in the kernel, so it involves the kernel.

Image does not involve the kernel.

Front 33

What is an orthonormal transformation?

Back 33

1. Preserves length
2. If put something with length one in, get length one out
3. All of its columns are linearly independent
4. All of its columns have length one.

Front 34

For an nxn matrix, how can you compare the determinants of A and A^T?

Back 34

det(A) = det(A^T)

Front 35

What do the columns of an orthogonal matrix form?

Back 35

An orthonormal basis of R^n

Front 36

Is the product of 2 orthogonal matrices also orthogonal?

Back 36

Yes

Front 37

Is the inverse of an orthogonal matrix orthogonal?

Back 37

Yes

Front 38

Is matrix A is nxm, what is a matrix A^T?

Back 38

mxn

Front 39

What is a square matrix?

Back 39

A = A^T

Front 40

What is a skew-symmetric matrix?

Back 40

-A = A^T

Front 41

What is the general form a of skew matrix?

Back 41

0 a b
-a 0 c
-b -c 0

Front 42

If a matrix is orthogonal, what is A x A^T?

Back 42

Identity Matrix

Front 43

Is rank(A) = rank(A^T)?

Back 43

Yes

Front 44

Is Ker(A) = Ker (AA^-1)?

Back 44

No, Ker(A) = Ker(AA^T)

Front 45

What does it mean if ker(A) = 0?

Back 45

AA^T is invertible

Front 46

What are the least squares solutions of Ax=b?

Back 46

x* = (A^TA)^-1 (A^T) (b)

(This is a matrix)

Front 47

What type of entity is the least squares solution?

Back 47

It is a matrix.

Front 48

Describe the determinant in words?

Back 48

The product of the values along the diagnoal.

Front 49

What is the form of a discrete trajectory?

Back 49

x(t+1) = Ax(t)

Front 50

What is the form of the characteristics equation?

Back 50

det(A-λI)

Front 51

What is the characteristic equatin of an n x n matrix?

Back 51

(-λ)^n + tr(A) (-λ)^(n-1) + ... + det(A) = 0

Front 52

What is trace?

Back 52

Sum of the diagonal entries.

Front 53

What is algebraic multiplicity?

Back 53

# of times a value is a root.

Front 54

Do A and A^T have the same eigenvalues?

Back 54

Yes

Front 55

Do A and A^ T have the same characteristic polynomials?

Back 55

Yes

Front 56

What does it mean that 2 matrices are similar?

Back 56

They have the same det and tr

Front 57

How do you find closed formula, (like lake pollutant quesiton)?

Back 57

1. x0 is the initial vector
a
b
c

2. x0 = c1v1 + c2v2 + ... + cnvn (Find eigenbasis to find c's)

3. x(t) = A^t x0 = c1 λ1^t v1 + ... + cn λn^t vt

4. Plug in values

Front 58

How do you know if an nxn matrix is diagonizable?

Back 58

If the dimensions of the iegnebasis add up to n

Front 59

What does this mean?:

lim A^t
t→∞

Back 59

Diagonalize A

Front 60

What is the relationsihp between A, S, and D?

Back 60

A = (S^-1) (D) (S)

Front 61

How can you find the closed formula solution?

Back 61

x(t) = A^t x0

1. Use D = (S^-1) (A) (S) to find D
2. A^t = (S^-1) (D^t) (S)
3. Since x(t) = (A^t) (x0), x(t) = (S^-1) (D^t) (S) )x0)

Front 62

When it says find closed formula, whta does it mean in other words?

Back 62

Find all entries of A^t.

Front 63

What is DeMoivre's formula?

Back 63

(cosΘ + i sinΘ)^n = cos(nΘ) + i sin(nΘ)

Front 64

If a 2 x 2 matrix has eigenvalues of a +ib and a - ib, what is S^-1 A S equal to?

Back 64

The matrix:
a -b
b a

Front 65

If z = a + bi, how do you find r and Θ?

Back 65

r = root (a^2 + b^2)

Θ = arctan (b/a)

Front 66

What do you do when you want to find some x(t + 1) = Ax(t) fraction as lim goes to infinity?

Back 66

Find eigenbasis, assuming λ = 1, and normalize it.

Front 67

When is A orthonogally diagonizable?

Back 67

If it is symmetric, (A^T= A)

Front 68

What are 2 ways to model evolution over time?

Back 68

Discrete and continuous cases.

Front 69

Describe a discrete case in words.

Back 69

It's like bank accounts tht compound 7% interest each year.

Front 70

Describe a discrete case algebraically.

Back 70

X(t+1) = 1.07 x(t)

Front 71

Describe a continuous case in words.

Back 71

It's like bank accounts with 7% annual interest, compounded continuously.

Front 72

Describe a continuous case algebrically.

Back 72

dx/dt = 0.07 x ~ x = x0 e^(0.07t)

Front 73

How do you find solutions to a discrete case, dx/dt = Ax, given A?

Back 73

1. Find the eigenvectors of A and designate this as S.
2. Plug λs and eignebasises into the equation x(t) = c1 e^(λ1t) v1 + ... + cn e^(λnt) vn
- You can solve for c's using x0
-The dominant term si associated with the larger eigenvalue, because e is raised to a larger exponent. It, therefore, determines the behavior of the system in the distance future.

Front 74

Describe the dominant term in a discrete case.

Back 74

the dominant term is associated with the larger eigenvalue, because e is raised to a larger exponent. It, therefore, determine teh behavior of the system in the distant future.

Front 75

Rewrite: e^(it)=cos(t) + i sint(t)

Back 75

e^(p+iq)t = e^(pt) (cos(qt) + i sin(qt))

Front 76

When is the continuous solution, dx/dt = Ax, a stable equilibrium solution?

Back 76

When all real parts of the eigenvalues are negative.

Front 77

What is the zero state of the discrete solution stable?

Back 77

tr(A) is less than 0
det(A) is greater than 0

Front 78

What do you do if you are asked to solve a discrete system, where A is a 2x2 matrix with complex eigenvalues?

Back 78

1. Subtract positive λ from the diagonal
2. Find the kernel of this matrix
3. Seperate into a [real coefficients] + i [complex coefficients]
- Designate S = [real coefficients complex coefficients]
4. Plug into x(t) = e^(pt) • S • the matrix:
cos(qt) -sin(qt)
sin(qt) cos(qt)
• S^(-1) • x0