Final Study Cards

Question 1

What is a scalar valued function?

Answer 1

A scalar valued function is a function which maps R^n onto R

Question 2

When is a scalar valued function linear?

Answer 2

A scalar valued function is linear if it has the form f(x1,…,xn) = a1x1 + a2x2 + ··· + anxn
i.e., it is affine with b = 0, or equivalently with f(0) = 0.

Question 3

When is a scalar valued function affline?

Answer 3

A scalar valued function is affline if it has the form f(x1,…,xn) = a1x1 + a2x2 + ···+ anxn + b for some numbers a1,…,an,b (so b = f(0)).

Question 4

What is the derivative matrix of a vector valued function?

Answer 4

The derivative matrix of a vector valued function f(x) = {f_1(x), f_2(x), f_3(x),…,f_n(x)
is the matrix such that
| f_1/dx_1 f_1/dx_2 f_1/dx_3 … f_1/dx_n|
| f_2/dx_1 f_2/dx_2 f_2/dx_3 … f_2/dx_n|
| f_3/dx_1 f_3/dx_2 f_3/dx_3 … f_3/dx_n|
| …
| f_n/dx_1 f_n/dx_2 f_n/dx_3 … f_n/dx_n|

f_1/dx_1 f_1/dx_2 f_1/dx_3 … f1/dx_n|

Question 5

What are the two linear approximations for a function F at x using the derivative matrix (where a is a close value)?

Answer 5

f(x) ≈f(a) + ((Df)(a))(x −a)
and
f(a + h) ≈f(a) + ((Df)(a)) h

Question 6

What is the linearity principle?

Answer 6

for c1,c2 ∈R and v1,v2 ∈R^2 we have f(c1v1 + c2v2) = c1f(v1) + c2f(v2).

Question 7

When is a function g linear?

Answer 7

When g(cx) = cg(x), g(x + y) = g(x) + g(y)

Question 8

What is the rotation matrix for R^2?

Answer 8

Aθ =
|cos θ −sin θ|
|sin θ cos θ|
.

Question 9

What is the identity matrix and what are it’s properties?

Answer 9

The identity matrix is an n x n square matrix such that its entries are equal to
|1 0 0 … 0|
|0 1 0 … 0|
|0 0 1 … 0|
|… |
|0 0 0 … 1|

The special property of the identity matrix is that any m x n matrix A multiplied by its respective n x n identity matrix is equal to A

so A*I_n = A

Question 10

What are the important properties of matrix multiplication

Answer 10

(MM1) It recovers matrix-vector multiplication: if A is an m ×n matrix, and x ∈ R_n is thought of as an n ×1 matrix, the matrix-matrix product Ax is the same as the matrix-vector product.
(MM2) A(B + C) = AB + AC and (A′ + B′)C′ = A′C′ + B′C′. (These “distributive laws” are the
reason we call it matrix multiplication.)
(MM3) A(BC) = (AB)C, and A(cB) = (cA)B = c(AB) for any scalar c. In particular, taking C
to be an m ×1 matrix that is a column vector v by another name, A(Bv) = (AB)v.
(MM4) If A is an m ×n matrix, then I_mA = A = AI_n, where I_m is the m × m identity matrix and I_n is the n ×n identity matrix.

Question 11

What are some important assertions about matrix multiplication

Answer 11

AB != BA
AB = AC does not imply B = C

Question 12

How does one set up and solve a markov chain problem?

Answer 12

First create a matrix M which models the total of each team after each cycle. For the n_th step in the cycle, the number at each position equals to M^n-1th * the original values

Question 13

What is the multivariable chain rule at a point = (v1,…,vn) ∈ R^n?

Answer 13

(D(f ◦ g))(v) = (Df)(g(v)) (Dg)(v)

Question 14

What is the definition of an inverse matrix B to a matrix A?

Answer 14

The inverse matrix B is a matrix such that BA = I_n and AB = I_n

Question 15

If A is invertible, what is Ax = b equal to?

Answer 15

x = (A^-1)(b)
It is important to note that the position of the A is maintained relative to the vector it is multiplying (so x != (b)(A^-1))

Question 16

What is true of n x n matrices with respect to their inverse?

Answer 16

If A and B are n × n matrices that satisfy AB = In then A is invertible and B is its
inverse; i.e., automatically the other equation BA = In holds.

Question 17

How can you immediately tell if a set of vectors is not linearly independent?

Answer 17

If it has a vector which is not non-zero

Question 18

When is a vector non-zero

Answer 18

When it has at least one entry which is not zero

Question 19

The upper left to lower right diagonal of the inverse of a square matrix is always what

Answer 19

1/The original value of the position in the diagonal (aka the reciprocal)

Question 20

What is the determinant of an upper or lower triangular matrix?

Answer 20

The product of its diagonal entries

Question 21

When is a 2x2 matrix inversible?

Answer 21

When it’s determinant is not equal to 0

Question 22

What is the determinant of a 2x2 matrix

Answer 22

The determinant is ad-bc where the slots are assigned
| a b |
| c d |

Question 23

When is a matrix A invertible?

Answer 23

A is invertible precisely when Ax = 0 has x = 0 as its only solution, and A is non-invertible precisely when Ax = 0 has a nonzero solution.

Question 24

What are some important matrix inversion rules?

Answer 24

(i) When A is invertible you can “cancel A” by multiplying both sides by A−1 (but there is a
caveat; see the Warning below):
– Cancelling an invertible matrix on the left: if AB = AC and A is invertible then
B = C. This holds because you multiply both sides on the left by A−1.
– Cancelling an invertible matrix on the right: if BA = CA and A is invertible then
B = C. For this you need to multiply both sides on the right by A−1.
– Warning: our caveat is that if AB = CA, then you cannot cancel A on the left on one
side and on the right on the other, so you cannot conclude in this case that B = C, even
when A is invertible (see Example 18.4.1 below).
(ii) If A and B are both invertible n ×n matrices then AB is also invertible, and (AB)−1 =
B−1A−1 (note the switch of order of multiplication on the right side!); see Example
18.4.3 below for an illustration.

Question 25

What is the inverse matrix of a 2x2 matrix (if it is invertible)?

Answer 25

It’s inverse matrix is

(1/ad-bc)|d -b|
|-c a|
where the original matrix is
|a b|
|c d|

Question 26

Newton Whatever Whatever Convergence Thing

Answer 26

Add It Maybe

Question 27

When is a collection of vectors considered linearly independent?

Answer 27

When none of the vectors belong to the span of the others

Question 28

When are a collection of vectors linearly independent

Answer 28

A collection of vectors v_1,…,v_k ∈R_n is linearly independent precisely when the
only collection of scalars a_1,…,a_k for which
a_1v_1 + a_2v_2 + ···+ a_kv_k = 0
is a_1 = 0,a_2 = 0,…,a_k = 0.

Question 29

How do you do the Gramm-Schmidt process for a collection of vectors v_1, v_2, …, v_n

Answer 29

Let v1,…,vk be nonzero n-vectors with span V in Rn.
Let w1 = v1 and define B1 to be {w1}(an orthogonal basis for V1!).
Let w2 = v2 −Projw1(v2)
Let w3 = v3 −Projw_2(v3) - Projw_2(v3)
and so on, defining at the jth step wj = vj − ProjVj−1(vj)

Question 30

How do you determine if a set of vectors {v1,v2,…,vk} are linearly independent?

Answer 30

If dim(span({v1,v2,v3}) = k

Question 31

What is the dimension of the orthogonal compliment to linear subspace v∈R_n?

Answer 31

The dimension of the orthogonal compliment is n - dim(V)

Question 32

What is the transpose of a matrix?

Answer 32

It is the resulting vector when you flip the values of the matrix across the upper left to lower right diagonal of the original matrix

Question 33

What does the transpose allow you to do?

Answer 33

The transpose allows you to move a matrix across a dot product, so for example

(Ax) dot y = x dot (A_Transpose(y))

Question 34

What are the important properties of matrix algebra

Answer 34

A(B + C) = AB + AC, (A + B)C = AC + BC, A(BC) = (AB)C, and
AIn = A = ImA for an m ×n matrix A. But AB 6= BA in general!
* (From before) Sometimes matrices are invertible. When they are invertible, you can multiply
by their inverse to cancel them: for example, if AB = AC and A is invertible then B = C,
whereas if AB = CA (with invertible A) then you can’t conclude anything.
* (From before) Invertible matrices are always square; i.e., n×n for some n. Inversion reverses
the order of multiplication: (AB)−1 = B−1A−1 if A and B are both invertible.
* (New) The transpose of an m×n matrix is an n×m matrix, and transpose reverses the order
of multiplication: (AB)> = B>A>.
* (New) If A is invertible so is A>, and (A>)−1 = (A−1)>.
* (New) For v,w ∈Rn viewed as n×1 matrices, the 1 ×1 matrix product v>w equals [v·w].
This yields efficient manipulation of dot products of many vectors at once via matrix algebra.

Question 35

What are the properties of a symmetric matrix

Answer 35

A_Transpose = A and the matrix is square (i.e. n x n). The inverse of a symmetric matrix is always symetric

Question 36

When is a given matrix A orthogonal?

Answer 36

When its transpose times it is equal to the identity matrix ((A_transpose)*A = I_ n)

or

The n columns of A are an orthonormal collection of m-vectors

Question 37

What is the inverse of an orthogonal matrix?

Answer 37

Its transpose

Question 38

Given matrices A and B are orthogonal, what is true of their product?

Answer 38

Their product regardless of order will be orthogonal

Question 39

What is the column space of a matrix?

Answer 39

The span of its columns

Question 40

When is an upper or lower triangular matrix invertible?

Answer 40

When all its diagonal entries are non-zero

Question 41

How does one solve an equation given the LU decomposition?

Answer 41

Say that L(Ux) = b and then say that Ux = y = {y1,y2,…,yn} and solve for Ly = b using back substitution.
Then replug in the values of Ux into the equation to solve via forward substitution

Question 42

How does one solve an equation given the QR decomposition?

Answer 42

Since Q is orthogonal we can say that
QRx = b => Rx = (Q_Transpose)b
Solve this to get the answer

Question 43

How do you solve the inverse of a matrix given its LU decomposition?

Answer 43

The inverse of a matrix given it’s LU decomposition is equal to L^-1U^-1
To solve these state that L’ and U’ are the inverses with the recipricol values along the diagonal and the rest as unknown of L and U, and then say that L’L = I and U’U = I solving for the unknowns using backwards or forwards substitution

Question 44

How do you solve the inverse of a matrix given its QR decomposition?

Answer 44

You know how to do it you twat

Question 45

How to find the QR decomposition of a matrix?

Answer 45

Given a matrix of form A = {v1,v2,v3,…,vn} run gramm-schmidt on the given vectors to get w1,w2,w3,…,wn. Q = the orthonormal form of these vectors

To find R write v1,…,vn as their equivalent form in terms of w1,..,wn
Multiply the coefficients of the given vn equation by the magnitude of wn is equivalent to the column vn