Flashcards in Raphael Deck (34)

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1

## linear combinations

###
Cu + Dv

for two vectors u and v, linear combinations are on a plane

if u and v are on the same line, the space of all the linear combinations is the line.

2

## Dot products

###
v= (v1, v2) w=(w1,w2)

v.w = v1w1 + v2w2

3

## Length of a vector

### sqrt(v.v) = sqrt(v1^2 + v2^2 + ...)

4

## Unit vector

###
length 1, u.u=1

we can divide any vector v by its length to get a unit vector in the same direction as v.

5

## Angle between a vector

###
a.b/|a||b| = cos theta

a.b=0 when v and w are perpendicular to each other

6

##
Cauchy Schwasz

Triangle Inequality

###
|v.w| <= ||v|| ||w||

||x +y|| <= ||x|| + ||y||

7

## Elimination

###
Multiplier = coefficient/pivot = Lij, where I is the row of coefficient and j is the column of the pivot

If the second pivot is zero then there is no solutions

solve x,y and z for 3 simultaneous equations

8

## Elimination matrix

### Identity matrix with a non-zero entry -L in position ij

9

## Matrix Pij for row exchange

### Identity matrix with column swapped so the rows will exchange

10

## Augmated matrix

###
Ax=b

[ A | b]

11

## Rules for matrix operations

###
- Can add matrices of the same size

- Multiply A and B if A is mxn and B is nxr

- (AB)C = A(BC)

- A( B+C) = AB + AC

- AB is not equal to BA

12

## Block matrix

### Can cut bigger matrices in to smaller blocks

13

## Inverse matrices( Properties)

###
A^(-1)A =I

Ax=b, x=A^(-1)b

(AB)^(-1) = B^-1A^-1

-The inverse exists if and only iff elimination produces n pivots for A nxn - full set

-The inverse is unique

-Ax=0, A does not have an inverse

14

## Calculating inverse matrices by elimination

###
Gauss- Jordan solves all three systems using the augmented matrix

1. Eliminate both sides to upper triangle form - circle pivots

2. Eliminate back up to a diagonal matrix of the left hand side of the augmented matrix

3. divide each row by the pivots so the identity matrix is on the LHS and A^(-1) on the RHS

The product of the pivots = determinants

15

## LDU decomposition

###
Elimination takes A to a upper triangle. (U for LU)

L is a lower triangle matrix, the identity matrix with the multiplies from elimination (l21, l31 ...) in there ij positions.

D is a diagonal matrix with the pivots on the diagonal

U - take the upper triangle matrix from elimination and divide each row by the pivots.

LDU = A

Using LU to solve Ax=b

sole Ly = b for y

solve Ux=y for x

16

## Inverse matrix for A is 2x2

###
A = [a b]

[c d]

A^-1 = 1/(ad -bc) [d -b]

[ -c a]

17

## Inverse of a diagonal matrix

###
A has d1,d2,d3 ...... on the diagonals and zeros elsewhere

A^-1 has 1/d1, 1/d2, 1/d3 on the diagonal and zeros elsewhere

18

## Transposes

###
Transfers the rows of a matrix A into a new matrix A^T

if A is a m x n matrix , A^T is a n x m matrix

(A+B)^T = A^T + B^T

(AB)^T = B^TA^T

(A^-1)^T = (A^T)^-1

19

## Symmetic matrices

###
S^T =S

(S^-1) = S

If S^T =S then LDU of S is LDL^T

20

## Permutation matrix

###
Has the columns of I in any order

P^-1 = P^T

21

##
Vector spaces

Real vector space

Subspace of a vector space

###
Vector space - The Space Rn consists of all the column vectors v with n components. E.g. R2 is the xy space, R3 is the xyz space and R1 is a line

Real vector space - Set of 'vectors' together with rules for vector addition and multiplication by real numbers

Subspace of a vector space - A set of vectors (including 0) such that if v and w are in the subspace

- v + w is in the subspace

- cv is in the subspace

22

## The column space of A

###
Consists of all linear combinations of the columns. These are the possible vectors Ax, denote C(A)

C(A) is a subspace of Rm

C(a^T) is a subset of Rn

23

## The Nullspace

###
N(A), consists of all the solutions to Ax=0. These solutions are vectors in Rn.

N(A) is a subspace of Rn

N(A^T) is a subset of Rm

24

## Special solutions

###
Tp describe N(A) as the span of some vectors, we need two special solutions.

1. Set free variables as 1 or 0 for each.

2. Solve u [x1,x2,x3,x4] =0

s1 = , s2=

3. N(u) = CS1 + DS2

If A is m x n, with m>n, there is at least one free variable and so at least one free special solution

Free columns have no pivots

25

## Reduced row echelon form

###
Continue to simplify a matrix after elimination to produce

1) Zeros above the pivots

2) 1's in the pivots

for m x n matrix

N(A)=N(U)=N(R)

if N(A) = 0

The column are independent, no combinations give 0 except the zero combination.

26

## The rank of A

###
Is the number of pivots. For rank r, A has r independent rows and r independent columns.

Rank is the dimension of C(A)

n - r is the dimension of N(A) for A is m x n

27

## The complete solution to Ax=b

###
X = Particular solution + (n-r) special solutions

For particular solution use the augmented matrix and set both free variable to 0

28

## Full column rank

###
Rank r = n

All columns have pivots

No free variables/ special solutions

N(A) = 0

If Ax=b, has a solution it is unique

29

## Full row rank

###
rank r = m

All rows have pivots, rref has no 0 rows

Ax=b has a solution for every b

C(A) = Rm

n-r =n-m, special solutions in N(A)

30