Coordinates, vector spaces and linear transformations

Question 1

Describe orthogonal vectors

Answer 1

2 vectors are orthogonal to each other if they are perpendicular, ie if their dot product = 0

Question 2

Define orthonormal

Answer 2

A set of vectors are orthonormal/form an orthonormal basis if all vectors have a magnitude of 1 and if they are all orthogonal to one another

Question 3

Give the 2x2 matrix that rotates a vector α degrees anti-clockwise

Answer 3

Question 4

Describe the determinant of a matrix product

Answer 4

For any square matrix, det(AB) = det(A) x det(B)

Question 5

Give the equation for the cofactor, c_ij of an element a_ij

Answer 5

Where M_ij (or minor) is the matrix left over when the elements in row i and column j have been removed

Question 6

Define the Jacobian

Answer 6

|J| (the determinant of the transformation matrix J) is the Jacobian, which gives the factor by which the volume element changes in a transformation.

Question 7

Describe 6 properties of determinants

Answer 7

1) Swapping rows and columns does not change the determinant
2) A determinant vanishes if one of the rows or columns contains only zeroes
3) If a row or column is multiplied by a constant, the determinant will also be multiplied by that constant.
4) A determinant vanishes if two rows or columns are multiples of each other
5) If we interchange a pair of rows or columns the determinant changes sign
6) Adding a multiple of one row or column to another doesn’t change the value of the determinant

Question 8

Describe the transpose of a matrix, A

Answer 8

A^T is obtained by switching the rows for columns. Starting with an m x n matrix, an n x m will be obtained.

Question 9

Describe the transposition of matrix products

Answer 9

Transposing a product of matrices reverses the order of multiplication, despite how many elements are involved. (ABC)^T = C^TB^TA^T

Question 10

Describe symmetrical matrices

Answer 10

If A^T = A, A is symmetric.

If A^T = -1, A is antisymmetric.

Question 11

Describe the product of orthogonal matrices

Answer 11

If A and B are orthogonal matrices, their product, C = AB will also be an orthogonal matrix

Question 12

Describe how to take the complex conjugate of a matrix

Answer 12

Take the complex conjugate of each element. (A*)_ij = a*_ij

Note: if A = A*, the matrix is real

Question 13

Describe hermitian conjugation

Answer 13

Also known as the Hermitian adjoint and represented by a superscript dagger (^†), it is a combination of complex conjugation and transposition, in either order: A^† = (A^T)* = (A*)^T

If A^† = A, A is Hermitian

If A^† = -A, A is anti-Hermitian

Question 14

Describe how to work out the Hermitian conjugate of a matrix product

Answer 14

(AB)^† = B^†A^†

Question 15

Describe the trace of a matrix

Answer 15

The trace of a matrix is the sum of all diagonal elements.

Tr(A) = a_¹¹ + a₂₂ + a₃₃ + ….

Question 16

Describe how to work out the determinant of a matrix A

Answer 16

Multiply each element of one row or column by its cofactor and add the results. Eg for a 3x3 matrix, the determinant can be calculated by working out a₁₁c₁₁ + a₁₂c₁₂ + a₁₃c₁₃

Question 17

Describe the inverse of a matrix A

Answer 17

The inverse of a matrix, defined as A^-1 is given such that AA^-1 = I

C is the matrix made up of the cofactors of A. C^T is known as the adjoint matrix to A, denoted by A^adj

Question 18

Describe the Kronecker delta, δ_ij

Answer 18

Question 19

Describe a unitary matrix

Answer 19

A square matrix U is unitary if U^†U = UU^† = I

Question 20

Describe linearly dependent vectors

Answer 20

A set of vectors X₁, X₂,…, X_n are linearly dependent if constants ci cant be found (not all zeroes) such that c₁X₁ + c₂X₂ + … + c_nX_n = 0. If no such constants exist, then X_i are linearly independent.

Question 21

Give the equation of a diagonalised matrix

Answer 21

D = L^-1ML

M is the diagonalisable matrix

L is the matrix whose columns are the eigenvectors of M

Question 22

Give the equation for the diagonalisable matrix M to any power

Answer 22

Mⁿ = LDⁿL^-1

Question 23

Describe a normal matrix

Answer 23

A matrix is described as normal if M^†M = MM^†

Question 24

Give the equation for the inverse of a matrix

Answer 24

Where C is the matrix made up of the cofactors of A. C is knowns as the adjoint matrix to A, A^adj

Question 25

State a property of all normal matrices

Answer 25

All normal matrices are all diagonalisable. Unitary, orthogonal, hermitian, and real symmetric matrices are all normal and hence diagonalisable

Question 26

Describe a hermitian matrix

Answer 26

A square matrix H is said to be hermitian if H^† = H

Question 27

Describe a real and symmetric matrix

Answer 27

A matrix is real and symmetric if its entries are real and S^T = S

Question 28

Describe an orthogonal matrix

Answer 28

A square matrix O is orthogonal if it has real entries and O^TO = OO^T = I

Question 29

Give the compact equation for finding the factor by which the volume element changes when we make a transformation.

Answer 29

dr’ = Jdr

Question 30

Describe how to test for linear dependence using determinants

Answer 30

The column vectors in question can be written as a matrix - columns forming columns. The determinant of this matrix M is taken.

If det(M) = 0, the vectors are linearly depedent

If det(M) ≠ 0, the vectors are linearly independent

Question 31

Give the equation needed to work out eigenvectors

Answer 31

Mv = λv

Question 32

Give the determinant of a 2x2 matrix A

Answer 32

DetA = a₁₁a₂₂-a₁₂a₂₁

Question 33

Give the characteristic equation used to find eigenvalues

Answer 33

Question 34

Define a diagonal matrix

Answer 34

A square matrix with elements only along the diagonal

Question 35

Describe Cramer’s rule

Answer 35