Lecture 9

Question 1

Vector space

Answer 1

Set V of vectors and a field F of scalars with addition and multiplication ⍺v∈V, u+v∈V
- associativity, commutativity, additive identity, additive inverse, associativity wrt scalar multiplication, distributivity wrt scalar/vector addition, scalar multiplication identity

Question 2

Linear function

Answer 2

f(u+v) = f(u) + f(v)
f(⍺u) = ⍺f(u)

Question 3

f(x) = |x|/x, f:R>R linear?

Answer 3

No

Question 4

f(x) = ax+b, f:R>R linear?

Answer 4

No

Question 5

Linear functions as matrix-vector multiplication ?

Answer 5

y = f(x) > y = Ax
A(u+v) = Au + Av
A(⍺u) = ⍺Au

Question 6

Shear operator

Answer 6

Change in angle:
1 b
a 1

Question 7

Rotation operator

Answer 7

cos(θ) -sin(θ)

sin(θ) cos(θ)

Question 8

Scale operator

Answer 8

a=x-direction, b=y-direction
a 0
0 b

Question 9

Reflection operator

Answer 9

-a:y-axis symmetry
-a 0
0 -b

Question 10

Translation

Answer 10

+ (a b)
1 0
0 1
nonlinear!

Question 11

Permutation matrix

Answer 11

permutation of the identity matrix (swap rows):
0 0 1
1 0 0
0 1 0

Question 12

Lower/Upper triangular matrix

Answer 12

lower: 0 j more than i, upper: 0 i more than j

Question 13

Rank

Answer 13

• number of linearly independent columns of the matrix

- rank(A) ≤ min(m,n), if = min(m,n) then full-rank otherwise rank deficient

Question 14

Singular matrix

Answer 14

A square matrix which is not invertible, that is det(A) = 0 (linearly independent row/columns) - if not singular then rank = n

Question 15

Sparse matrix

Answer 15

O(min(n,m)) non-zero entries

Question 16

A+B number of operations dense vs sparse matrices

Answer 16

O(n^2) vs

O(nnz(A)+nnz(B))

Question 17

Dense representation (DNS)

Answer 17

AA = [#rows #col v1 v2 v3…]

stores zeros, row-wise..

Question 18

Coordinate representation (COO)

Answer 18

data = [v1 v2 v3...]
row = [rowv1, rowv2,...]
col = [colv1, colv2...]
nnz*doubles + 2nnz*integers
no zeros and not sorted

Question 19

COO how many bytes? (32 bits integer, 64 bits float, 1 byte = 8 bits)
0    0    1.3
-1.5 0.2  0
5    0     0
0    0.3  3
0    0     0

Answer 19

nnz64+2nnz32 = 96 bytes

Question 20

Compressed Sparse Row representation (CSR)

Answer 20

data = [v1, v2,…] (nnz integers)
col = [colv1, colv2,…] (nnz doubles)
rowptr = [i1, i2,.., in, nnz] (size n+1)
i1 is the starting index in data of row1 (row offset) - always 0!
fast operations between sparse matrices / matrix-vector product.

Question 21

CSR representation (rowptr)
1   5  0  0   1
0  2  3  0  0
0  0  0  0  0
0  0  0  4  0
0  0  6  9  7

Answer 21

rowptr = [0 3 5 5 6 9]

Question 22

Inverse of a permutation matrix - nnz

Answer 22

its transpose, nnz = n

Question 23

Matrix in block form / block diagonal

Answer 23

matrix partitioned into blocks (sub matrices)
M =
A B
C D
block diagonal if off-diagonal matrices are zero matrices (C and B here)

Question 24

A rotates θ, what rotates -θ?

Answer 24

A.T, A^-1

Question 25

How many solutions Ax = b if A is non-singular?

Answer 25

1 solution, x=A^-1 b

Question 26

How many solutions Ax = b if A is singular?

Answer 26

It depends on b

Question 27

from/to CSR python

Answer 27

import scipy as sp
import scipy.sparse

csr = sp.sparse.csr_matrix(A)
print(csr.indptr)

sp.sparse.csr_matrix( (data,indices,indptr), shape=(x,y) ).todense()