Coordinates, vector spaces and linear transformations Flashcards Preview

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Flashcards in Coordinates, vector spaces and linear transformations Deck (45)
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1

Describe orthogonal vectors

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

2

Define orthonormal

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

3

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

4

Describe the determinant of a matrix product

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

5

Give the equation for the cofactor, cij of an element aij

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

6

Define the Jacobian

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

7

Describe 6 properties of determinants

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

8

Describe the transpose of a matrix, A

Ais obtained by switching the rows for columns. Starting with an m x n matrix, an n x m will be obtained. 

9

Describe the transposition of matrix products

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

10

Describe symmetrical matrices

If AT = A, A is symmetric. 

If AT = -1, A is antisymmetric.

11

Describe the product of orthogonal matrices

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

12

Describe how to take the complex conjugate of a matrix

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

Note: if A = A*, the matrix is real 

13

Describe hermitian conjugation

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 = (AT)* = (A*)T

 

If A = A, A is Hermitian

If A = -A, A is anti-Hermitian

 

 

14

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

(AB) = BA

15

Describe the trace of a matrix

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

Tr(A) = a11 + a22 + a33 + .... 

16

Describe how to work out the determinant of a matrix A

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

17

Describe the inverse of a matrix A

 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. Cis known as the adjoint matrix to A, denoted by Aadj

18

Describe the Kronecker delta, δij

19

Describe a unitary matrix

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

 

20

Describe linearly dependent vectors 

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

21

Give the equation of a diagonalised matrix 

D = L-1ML

M is the diagonalisable matrix 

L is the matrix whose columns are the eigenvectors of M 

22

Give the equation for the diagonalisable matrix M to any power 

Mn = LDnL-1

23

Describe a normal matrix

 A matrix is described as normal if MM = MM

24

Give the equation for the inverse of a matrix

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

25

State a property of all normal matrices 

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

26

Describe a hermitian matrix

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

27

Describe a real and symmetric matrix

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

28

Describe an orthogonal matrix 

A square matrix O is orthogonal if it has real entries and OTO = OOT = I

29

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

dr' = Jdr 

30

Describe how to test for linear dependence using determinants 

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

31

Give the equation needed to work out eigenvectors

Mv = λv

32

Give the determinant of a 2x2 matrix A

DetA = a11a22-a12a21

33

Give the characteristic equation used to find eigenvalues

34

Define a diagonal matrix

 A square matrix with elements only along the diagonal

35

Describe Cramer's rule 

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