Alternate Coordinate Systems

Question 1

Define:

Orthogonal complement of a subspace

Orthogonal complements 00:00

Answer 1

V is a subspace of Rⁿ and the orthogonal complement of V is
a subset of Rⁿ where every member is orthogonal to every member of V
V^⟂={x ∈ Rⁿ| x.V=0 for every V∈ Rⁿ}

Note: dot product of orthogonal vectors is 0

Question 2

Is the orthogonal complement of subspace V a subspace?

Orthogonal complements 01:58

Answer 2

Yes

Question 3

What is the row space of matrix A equal to?

Orthogonal Complement 6:33

Answer 3

C(A^T)
(column space of transpose of A)

Question 4

What is the purpose of reduced row echelon form?

External

Answer 4

Reduced row echelon form is a type of matrix used to solve systems of linear equations. such as finding out if two vectors are linearly independent

Question 5

What is the coordinates of a with respect to basis of the subspace V?
Assume:
basis V={v1,v2,…,vk}
a=c1v1+c2v2+…+ckvk
V is a subspace of Rⁿ

Coordinates with respect to a basis 00:00

Answer 5

coordinates of a with respect to basis of V, is: [c1,c2,…,ck]
Note: the coefficients can be 0, so dim(a) can be less than k (dim(subspace V))

Question 6

How can we change back the coordinates of A with respect to basis of V sub space to its original coordinates?

Coordinates with respect to a basis 2:42

Answer 6

coordinates of A with respect to V basis: it’s a linear combination of the basis: a1v1+a2v2+…+akvk, so
A_{with respect to V basis}=[a1,a2,…,ak]
A’s standard presentation is: A_{standard basis}=a1v1+a2v2+…+akvk
therefore, in order to restore the original form, we do this:
A_{standard basis}= C A_{with respect to V basis}
C_n*k=[v1 v2 … vk] (n*k matrix that contains basis vectors as its columns, aka. change basis matrix)

Note: Change basis matrix is NOT the same as the transformation matrix. Change of basis formula relates coordinates of one and the same vector in two different bases, whereas a linear transformation relates coordinates of two different vectors in the same basis.

Question 7

If we have 2 linearly independent vectors in R² then these two can be basis for R². True/False

Invertible change of basis matrix 06:48

Answer 7

True

Question 8

If B is a basis of sub space V of Rⁿ, and if the number of elements in the basis is n, then B is a basis for Rⁿ. True/False

Invertible change of basis matrix 00:00

Answer 8

True

.
In this case, C (n*n matrix containing basis as columns) is invertible, therefore
C^-1C A_{with respect to V basis}=C^-1A_original
Then
A_{with respect to V basis}=C^-1A_original

Question 9

Sometimes finding the transformation matrix A, is not an easy task, what can be done about it? Give an example

Answer 9

We can define a new coordinate system using a new basis set. using eigenvectors is a way of doing this.

Changing coordinate systems to help find a transformation matrix 04:26

Showing that an eigenbasis makes for good coordinate systems

Question 10

What is the relation between

vector X original coordinates and,
1) its matrix transformation,
2) its change basis matrix transformation

.
Transformed X_{standard coord} and Transformed X_{change basis coord}

Changing coordinate systems to help find a transformation matrix 06:35

Answer 10

AX=T(X) [matrix transformation]
X_Org=CX_B [C= change basis matrix]
X_BD=[T(X)]_B
T(X)=C [T(X)]_B
since the transformation matrix, is a vector with respect to the orginial coordinates

Question 11

What do we need to do to convert from the standard basis (B) to the basis given by the eigenvectors (B′)?

Answer 11

.
To convert from the standard basis (B) to the basis given by the eigenvectorrs (B′), we multiply by the inverse of the eigenvector matrix V⁻¹. Since the eigenvector matrix V is orthogonal, V^T=V⁻¹

Source
Calculating PCA using eigenvector basis change