Linear Algebra Octave

Question 1

Initialize a vector

Answer 1

V1 = [1 2 3]

Question 2

Addition of two vectors

Answer 2

v1 + v2

Question 3

Transpose of a vector

Answer 3

1. v’ (hermitian)
2. v.’ - regular transpose
3. transpose(v) - regular transpose

Question 4

Scalar multiplication

Answer 4

2 * v

Question 5

Dot product approaches

Answer 5

1. dp = sum (v1 .* v2)
2. dp = dot(v1, v2)
3. dp = v1T * v2 (v1 and v2 must be column vectors)

Question 6

Create one random number

Answer 6

randn / randn()

Question 7

Create random vector of size 10

Answer 7

v = randn(10, 1)

Question 8

Create a matrix of size 4 X 6

Answer 8

mat = randn(4, 6)

Question 9

Get the size of the matrix

Answer 9

size(mat) = (4, 6)

Question 10

Just get the number of rows of a matrix

Answer 10

size(mat, 1) = 4

Question 11

Just get the number of columns of a matrix

Answer 11

size(mat, 2) = 6 ; length(mat) = 6

Question 12

Get the first column of a matrix

Answer 12

mat(:, 1)

Question 13

Get the first and third columns of a matrix

Answer 13

mat(:, [1,3])

Question 14

Get the one to third columns of a matrix

Answer 14

mat(:, [1:3])

Question 15

Repeat 1st column twice in a matrix

Answer 15

mat(:, [1,1])

Question 16

Get the first row of a matrix

Answer 16

mat(1, :)

Question 17

Get the first and third rows of a matrix

Answer 17

mat([1,3], :)

Question 18

Get 1 to third rows in a matrix

Answer 18

mat([1:3], :)

Question 19

Repeat 1st row twice in a matrix

Answer 19

mat([1,1], :)

Question 20

Repeat 1st row n times in a matrix

Answer 20

mat(repelem(1,n), :)

Question 21

Create a zero vector of size 4

Answer 21

zero_vector = zeros(4,1)

Question 22

magnitude of a vector

Answer 22

1. sqrt(dot(v,v))
2. sqrt(sum(v .* v))
3. norm(v)

Question 23

Angle between two vectors

Answer 23

theta = acos(dot(v1, v2)/(norm(v1) * norm(v2)))

Question 24

How to know if two vectors are orthogonal

Answer 24

If the dot product of the two vectors is 0

Question 25

How to create just positive random number

Answer 25

rand

Question 26

create duplicate dependent vector from v1

Answer 26

v2 = randn/rand * v1

Question 27

absolute value of a number

Answer 27

abs(-1)

Question 28

Perform hadamard multiplication

Answer 28

1. v1 . * v2

Question 29

What does v1 * v2 represent

Answer 29

matrix multiplication

Question 30

Outer product

Answer 30

v .* w’

Question 31

cross product of vectors

Answer 31

cross(v, w)

Question 32

create complex number

Answer 32

complex(3, 4) = 3+4i

Question 33

Create complex vector

Answer 33

v = [3 4i complex(5,2) 2-5i]

Question 34

transpose(3 + 4i) =

Answer 34

3 + 4i

Question 35

Hermitian transpose of v = 3+4i - write syntax

Answer 35

v’ * v = 25

Question 36

Unit vectors - mu =

Answer 36

1/norm(v)

Question 37

Normalize a vector

Answer 37

mu * v = norm_vector such that norm(norm_vector) = 1

Question 38

Creating an identity matrix

Answer 38

m = eye(5). Can create rectangular matrices as well. 1’s is equal to number of columns. But identity matrices are only square matrices in practice

Question 39

Creating an zeros matrix

Answer 39

m = zeros(4). Can create rectangular matrices as well

Question 40

creating diagonal matrix

Answer 40

m = diag([1 2 3 4 5]); Can create rectangular matrices as well

Question 41

Creating upper triangular matrix

Answer 41

s = randn(5)
ut = triu(s); Can create rectangular matrices as well

Question 42

Creating lower triangular matrix

Answer 42

s = randn(5)
lt = tril(s); Can create rectangular matrices as well

Question 43

Concatenate two matrices

Answer 43

m1 = randn(5* 3)
m2 = randn(5 * 5)
C = [m1 m2]

Question 44

Addition of matrices

Answer 44

A+B ; both are of same dimensions

Question 45

Shifting a matrix

Answer 45

D = randn(10)
shft = D * (0.5 * eye(10))

Question 46

Two applications of diag keyword

Answer 46

1. diag(matrix) =pulls out the diagnoal as a row vector, [1;2;3;4;5;6]
2. diag(vector) = creates a diagonal matrix with the vector elements as diagonal.

Question 47

Calculating trace

Answer 47

1. trace(matrix)
2. sum(diag(matrix))

Question 48

Properties that prove trace is a linear operator

Answer 48

1. trace(a+b) = trace(a) + trace(b)
2. l * trace(a) = trace(l*a)
   closed under both addition and subtraction

Question 49

Create a matrix with values from 1 to 12 and shape 3X4

Answer 49

A = reshape(1:12, 3,4)

Question 50

Broadcasting three ways

Answer 50

a =reshape(1:12, 3,4)
r = [1 2 3 4]
c = [100 200 300]’

1. a + repmat(r, size(a,1), 1)
   a + repmat(c, 1, size(a,2)))
2. bsxfun(@plus, a, r)
   bsxfun(@plus, a, c)
3. a + r
   a + c

Question 51

bsxfun stands for

Answer 51

binary expansion

Question 52

Creating a symmetric matrix from a non-symmetric matrix a

Answer 52

s = (a * a’)/m^2

Question 53

Hadamard matrix multiplication

Answer 53

.* (* for regular multiplication)

Question 54

Hadamard multiplication rule

Answer 54

Both the matrices must be of same size