School of Computing Theory Assignment 1 of 4

COMP3670/6670: Introduction to Machine Learning

Maximum credit. 100

Exercise 1 Solving Linear Systems (4+4 credits)

Find the set S of all solutions x of the following inhomogenous linear systems Ax = b, where A and b

are defined as follows. Write the solution space S in parametric form.

(a)

A =

0 1 5

1 4 3

2 7 1

, b =

−4

−2

−2

(b)

A =

2 3 1

4 0 3

, b =

6

12

Exercise 2 Inverses (4 credits)

For what values of [a, b, c]

T ∈ R

3 does the inverse of the following matrix exist?

1 a b

1 1 c

1 1 1

Exercise 3 Subspaces (3+3+3+3 credits)

Which of the following sets are subspaces of R

3

? Prove your answer. (That is, if it is a subspace, you

must demonstrate the subspace axioms are satisfied, and if it is not a subspace, you must show which

axiom fails.)

(a) A = {(x, y) ∈ R

2

: x ≥ 0, y ≥ 0}

(b) B = {(x, y, z) : x + y + z = 0}.

(c) C = {(x, y) ∈ R

2

: x = 0 or y = 0}

(d) D = The set of all solutions x to the matrix equation Ax = b, for some matrix A and some vector

b. (Hint: Your answer may depend on A and b.)

Exercise 4 Linear Independence (4+8+8 credits)

Let V and W be vector spaces. Let T : V → W be a linear transformation.

(a) Prove that T(0) = 0.

(b) For any integer n ≥ 1, prove that given a set of vectors {v1, . . . vn} in V and a set of coefficients

{c1, . . . , cn} in R, that

T(c1v1 + . . . + cnvn) = c1T(v1) + . . . + cnT(vn)

(c) Let {v1, . . . vn} be a set of linearly dependent vectors in V .

Define w1 := T(v1), . . . , wn := T(vn).

Prove that {w1, . . . , wn} is a set of linearly dependent vectors in W.

Exercise 5 Inner Products (4+8 credits)

(a) Show that if an inner product h·, ·i is symmetric and linear in the first argument, then it is bilinear.

(b) Define h·, ·i for all x = [x1, x2]

T ∈ R

2 and y = [y1, y2]

T ∈ R

2 as

hx, yi = x1y1 + x2y2 + 2(x1y2 + x2y1)

Which of the three inner product axioms does h·, ·i satisfy?

Exercise 6 Orthogonality (8+6 credits)

Let V denote a vector space together with an inner product h·, ·i : V × V → R.

Let x, y be non-zero vectors in V .

(a) Prove or disprove that if x and y are orthogonal, then they are linearly independent.

(b) Prove or disprove that if x and y are linearly independent, then they are orthogonal.

Exercise 7 Properties of Norms (4+4+10 credits)

Given a vector space V with two norms k · ka : V → R≥0 and k · kb : V → R≥0, we say that the two

norms k · ka and k · kb are ε-equivalent if for any v ∈ V , we have that

εkvka ≤ kvkb ≤

1

ε

kvka.

where ε ∈ (0, 1].

If k · ka is ε-equivalent to k · kb, we denote this as k · ka

ε∼ k · kb.

(a) Is ε-equivalence reflexive for all ε ∈ (0, 1]?

(Is it true that k · ka

ε∼ k · ka?)

(b) Is ε-equivalence symmetric for all ε ∈ (0, 1]?

(Does k · ka

ε∼ k · kb imply k · kb

ε∼ k · ka?)

(c) Assuming that V = R

2

, prove that k · k1

ε∼ k · k2. for the largest ε possible.

Exercise 8 Projections (3+3+3+3 credits)

Consider the Euclidean vector space R

3 with the dot product. A subspace U ⊂ R

3 and vector x ∈ R

3 are

given by

U = span

1

1

1

,

2

1

0

, x =

12

12

18

(a) Show that x 6∈ U.

(b) Determine the orthogonal projection of x onto U, denoted πU (x).

(c) Show that πU (x) can be written as a linear combination of [1, 1, 1]T and [2, 1, 0]T

.

(d) Determine the distance d(x, U) := miny∈U kx − yk2.

COMP3670/6670

# Assignment 1 of 4 Exercise 1 Solving Linear Systems

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