Midterm 2 Review

BFITZ - Spring 2024

DONT FORGET THIS!!

f(t) = a0 + a1t

What is the error of least squares

E = ||b-Ax||²

What do least squares approximation solutions satisfy?

Ax(hat) = Pb

What is the least squares problem of Ax=b?

A^TAxhat = A^Tb

DONT FORGET THIS!!

V is big is easier to project

What does the sum of the projection matrix onto V and the projection matrix into the orthogonal space of V equal to…

the identity matrix

Fill in the blank

n (dimension)

Remember how to form projection matrix when dim(V) = …?

dim(V) = 1

trace(P) = ?

dim(V)

If v belongs to the orthogonal vector space of V, then Pv = ?

The zero vector

If a vector v belongs to the V vector space, then Pv = ?

The vector v

Projection matrices are ___ and ____

symmetric and idempotent

What is the projection matrix formula?

P = X(X^TX)^-1X^T

Fill in the blank

m - r

Which subspaces are orthogonal components?

Null(A) and Col(A^T),
Null(A^T) and Col(A)

How can the Euler characteristic be written in Betti numbers?

X(G) = h0(G) - h1(G)

What is the first Betti number equal to?

h1(G) = circuit rank = dim Null(A)

What is the zeroth Betti number equal to?

h0(G) = # of connected compoents = dim Null(A^T)

What is the dimension of the eigenspace of lambda?

the geometric multiplicity of lambda

What is dim Null(A^T)?

nullity(A^T)

What is dim Null(A)?

nullity(A)

What is Col(A^T)?

rank(A^T) = rank(A)

What is dim Col(A)?

rank(A)

How do you form the basis for Null(A^T) of a digraph?

Form classification vectors denoting which nodes are in each connected component. These form the basis for the digraph.

How do you form the basis of Null(A) for a digraph?

Remove the minimum number of arrows to break a cycle, and make cycle vectors for each additional vector that was purged. These vectors are the basis for Null(A).

What vectors form the basis of A^T?

Pivot solutions of A^Tv = 0

What vectors form the basis of Null(A)?

The pivot solutions of Av = 0

What vectors form the basis of Col(A^T)?

The pivot columns of A^T and nonzero rows of rref(A)

What is the basis of Col(A)

Span{pivot cols of A} and the Span{nonzero rows of rref(A^T)}

What is a basis of a vector space?

A set of vectors that are linearly independent and whose linear combination could be any vector in the vector space

Vectors are independent if their rref matrix where the vectors are columns is ___

full column rank and has nullity = 0

How can you determine independence for vectors geometrically?

If the vectors are not multiples of each other, or do not sum to zero, then they are independent

What is the linear independence test?

The columns of A are independent if nullity(A)=0

What is the weight condition for linear independence?

For the sum to be the zero vector, all of the weights must be 0

What is the weight condition for linear dependence?

At least one of the weights is not equal to 0

Remember Col(A) to Null(B) conversion

Col(A) = Null(B)

Remember Null(A) to Col(B) conversion

Null(A) = Col(B) example

What are the conditions for a vector space?

The zero vector must be in it, AND any vector in the space can be shown as a linear combination of any other vectors in the space

It is generally easier to produce vectors in ____ spaces and harder to produce vectors in ____ spaces?

Easier: Col(A) and Col(A^T). Harder: Null(A) and Null(A^T)

Col(A^T) is called ___ and Null(A^T) is called ___?

row space and left null space

V is in the span of vectors if augmenting a matrix where the vector in the span are the columns with v is consistent

How do you verify if there is a weight configuration vector w in which Aw = v?

check if v is in Col(A)

How do you produce a vector in Col(A)

take any linear combination of the columns of A. That vector will be in Col(A)

How do you verify if a vector v is in Col(A)

Augment A with v and solve the system. If the system is consistent, then v is in Col(A)

From this definition, vectors in Col(A) have the same number of coordinates as the number of ____ of A

the number of rows of a

for a eigenvector Aw = (lambda)w, the output Aw is _____ to the input w

The output is parallel to the input

The eigenspace of A associated to an eigenvalue lambda is equal to…?

Null(XA(lambda))

How can you verify if a path p can be considered a cycle vector?

only if p is in Null(A)

How do you verify a vector v is in Null(A)

Multiply the vector v by A. If the product is the null vector, then Null(A)

How do you generate the Null(A)?

Augment A with the O vector and solve system. The linear combinations of solution vector(s) are Null(A)

By this definition, a vector in the null space of A has how many coordinates equal to the number of ____ of A

columns of A

what are the lower and upper bounds for the sums of the gms for each eigenvalue of an n x n matrix A

lower bound: the number of eigenvalues of A

upper bound: the size of A

The geometric multiplicity of lambda is gmA(lambda) =

the nullity of the characteristic matrix XA(lambda)

the eigenvalues of a triangular matrix are...

the diagonal entries

if 0 is not an eigenvalue of A, what can we say about A

A is nonsingular

What can we say about A?

A is singular

the scalar lambda is an eigenvalue of an n x n matrix A if and only if (express in rank and nullity)

rank(XA(lambda)) < n and nullity(XA(lambda)) > 0

when is the lambda an eigenvalue of A

when the characteristic matrix is singular