Raphael

Question 1

linear combinations

Answer 1

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.

Question 2

Dot products

Answer 2

v= (v1, v2)   w=(w1,w2)
v.w = v1w1 + v2w2

Question 3

Length of a vector

Answer 3

sqrt(v.v) = sqrt(v1^2 + v2^2 + …)

Question 4

Unit vector

Answer 4

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.

Question 5

Angle between a vector

Answer 5

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

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

Question 6

Cauchy Schwasz

Triangle Inequality

Answer 6

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

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

Question 7

Elimination

Answer 7

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

Question 8

Elimination matrix

Answer 8

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

Question 9

Matrix Pij for row exchange

Answer 9

Identity matrix with column swapped so the rows will exchange

Question 10

Augmated matrix

Answer 10

Ax=b

[ A | b]

Question 11

Rules for matrix operations

Answer 11

• 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

Question 12

Block matrix

Answer 12

Can cut bigger matrices in to smaller blocks

Question 13

Inverse matrices( Properties)

Answer 13

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

Question 14

Calculating inverse matrices by elimination

Answer 14

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

Question 15

LDU decomposition

Answer 15

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

Question 16

Inverse matrix for A is 2x2

Answer 16

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

Question 17

Inverse of a diagonal matrix

Answer 17

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

Question 18

Transposes

Answer 18

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

Question 19

Symmetic matrices

Answer 19

S^T =S
(S^-1) = S
If S^T =S then LDU of S is LDL^T

Question 20

Permutation matrix

Answer 20

Has the columns of I in any order

P^-1 = P^T

Question 21

Vector spaces
Real vector space
Subspace of a vector space

Answer 21

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

Question 22

The column space of A

Answer 22

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

Question 23

The Nullspace

Answer 23

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

Question 24

Special solutions

Answer 24

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

Question 25

Reduced row echelon form

Answer 25

``` 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. ```

Question 26

The rank of A

Answer 26

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

Question 27

The complete solution to Ax=b

Answer 27

X = Particular solution + (n-r) special solutions For particular solution use the augmented matrix and set both free variable to 0

Question 28

Full column rank

Answer 28

``` Rank r = n All columns have pivots No free variables/ special solutions N(A) = 0 If Ax=b, has a solution it is unique ```

Question 29

Full row rank

Answer 29

``` 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) ```

Question 30

Linear independence

Answer 30

The column of Matrix A are linearly independent when the only solution to Ax=0 is x=0 The sequence of vectors v1, v2, v3 ..... vn is linearly independent if the only combination x1v1 + x2v2 ....+xnvn =0 is when xi=0 The column are independent exactly when r=n. There are n pivots, no free variable N(A) = 0 If mm.

Question 31

A set of vectors span a space

Answer 31

if their linear combinations fill the space. | The columns of A span C(A), they may be dependant

Question 32

The row space of a matrix

Answer 32

is the subspace of Rn spanned by the rows. | The row space is C(A^T)

Question 33

A basis for a vector

Answer 33

Is a sequence of vectors with 1) The basis vectors are linearly independent 2) The basis vectors span the space Basis for C(A) is the pivot columns Basis for C(A^T) is the pivot rows

Question 34

Dimensions

Answer 34

``` Dimensions of a space is the number of vectors in every basis For a m x n matrix with rank r C(A) and C(A transpose) - dimension = r N(A) - dimension = n-r N(A transpose) - dimension = m-r ``` In R^n : C(R^T) and N(R) dimensions add to n in R^m: C9R) and N(R^T) dimensions add to m C(A^T) = C(R^T) C(A) is not equal to C(R) dim(C(A)) + dim(N(A)) = n N(A) =N(R)