Linear Transformation And Linear System

Question 1

Affine transformation

Answer 1

An affine transform is a combination of a linear transform with a translation

Question 2

Affine Transform - Formal Definition

Answer 2

Question 3

Householder reflection matrix

Answer 3

Question 4

Anisotropic

Answer 4

When the scaling factors across different dimensions are different, the scaling is said to be anisotropic

Question 5

All linear transformations defined by matrix multiplication can be expressed as a sequence of

Answer 5

Rotations/reflections, together with a single anisotropic scaling

Question 6

Dimensionality of a vector space

Answer 6

The number of members in every possible basis set of a vector space V is always the same. This value is referred to as the dimensionality of the vector space.

Question 7

Matrix Invertibility and Linear Independence

Answer 7

An n × n square matrix A has linearly independent columns/rows if and only if it is invertible

Question 8

The normal equation

Answer 8

Question 9

Left-inverse of the matrix A in the normal equation

Answer 9

Question 10

Right-inverse of the matrix A in the normal equation

Answer 10

Question 11

Converting coordinates from one system to another

Answer 11

• A is the original coordinates system
• B is the target coordinates system
• x_b and x_a and the vectors being transformed

Question 12

The normal equation expressed as a change of coordinates system

Answer 12

Question 13

Disjoint Vector Spaces

Answer 13

Two vector spaces U⊆Rⁿ and W⊆Rⁿ are disjoint if and only if the two spaces do not contain any vector in common other than the zero vector

Question 14

Orthogonal Vector Spaces

Answer 14

Two vector spaces U⊆Rⁿ and W ⊆ Rⁿ are orthogonal if and only if for any pair of vectors u ∈ U and w ∈ W, the dot product of the two vectors is 0

Question 15

Rectangular matrices are said to be of full rank when either the rows or the columns are linearly independent

Answer 15

The former is referred to as full row rank, whereas the latter is referred to as full column rank

Question 16

Left Null Space

Answer 16

The left null space of an n × d matrix A is the subspace of Rn containing all column vectors x ∈ Rn, such that AT x = 0. The left null space of A is the orthogonal complementary subspace of the column space of A

Question 17

The four fundamental subspaces of linear algebra

Answer 17

Question 18

Linear Independence and Gram Matrix

Answer 18

The matrix A^T A is said to be the Gram matrix of the column space of an n × d matrix A. The columns of the matrix A are linearly independent if and only if A^T A is invertible

Question 19

Orthogonality of Basis Vectors for the discrete cosine transform

Answer 19

The dot product of any pair of basis vectors b_p and b_q of the discrete cosine transform for p != q is 0

Question 21

Regularized left and right inverse

Answer 21

Question 22

Orthogonal projection of a point onto a line

Answer 22

Question 23

Projection of a point onto a line

Answer 23

the Subspace goes in the Subscript

Question 24

Moore-Penrose pseudoinverse

Answer 24

Question 25

Moore-Penrose pseudoinverse relation to other inverses

Answer 25

• The conventional inverse, the left-inverse, and the right-inverse are special cases of the Moore-Penrose pseudoinverse.
• When the matrix A is invertible, all four inverses are the same.
• When only the columns of A are linearly independent, the Moore-Penrose pseudoinverse is the left-inverse.
• When only the rows of A are linearly independent, the Moore-Penrose pseudoinverse is the right-inverse.
• When neither the rows nor columns of A are linearly independent, the Moore-Penrose pseudoinverse provides a generalized inverse that none of these special cases can provide.
• Therefore, the Moore-Penrose pseudoinverse respects both the best-fit and the conciseness criteria like the left- and right inverses.

Question 26

Orthonormal matrix relation to the identity matrix

Answer 26

Question 27

Projection matrix in the normal equation

Answer 27

Question 28

Computing the projection matrix via QR decomposition

Answer 28

Question 29

Condition Number

Answer 29

Let A be a d×d invertible matrix. Let ||Ax||/||x||
be the scaling ratio of vector x. Then, the condition number of A is defined as the ratio of
the largest scaling ratio of A (over all d-dimensional vectors) to the smallest scaling ratio
over all d-dimensional vectors.

Question 30

Inner Products: Restricted Definition

Answer 30

A mapping from x, y ∈ Rⁿ to x, y ∈ R is an inner product if and only if 〈x, y〉 is always equal to the dot product between Ax and Ay for some n × n non-singular matrix A. The inner product 〈x, y〉 can also be expressed using the Gram matrix S = A^T A

Question 31

Inner product: cosines and distances

Answer 31

When the linear transformation A is a rotreflection matrix, the matrix S is the identity matrix, and the inner product specializes to the normal dot product

Question 32

Inner-Product: General Definition

Answer 32

The real value 〈u, v 〉 is an inner product between u and v, if it satisfies the following axioms for all u and v:

Question 33

Complex-valued inner product

Answer 33

Question 34

Conjugate Transpose of Vector and Matrix

Answer 34

The conjugate transpose v^∗ of a complex vector v is obtained by transposing the vector and replacing each entry with its complex conjugate. The conjugate transpose V^∗ of a complex matrix V is obtained by transposing the matrix and replacing each entry with its complex conjugate

Question 35

Answer 35

Question 36

Orthogonality in Cⁿ

Answer 36

Question 37

Orthogonal Matrix with Complex Entries

Answer 37

Question 38

The Discrete Fourier Transform

Answer 38

Used for finding an orthonormal basis for time-series in the complex domain