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COMP3670/6670: Introduction to Machine Learning

Release Date: 3 Aug 2022

Due Date: 23:59pm, 28 Aug 2022

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.

Exercise 2 Inverses (4 credits)

Find the inverse of the following matrix, if an inverse exists.1 1 22 3 1

3 4 2

Exercise 3 Subspaces (3+3+3+4 credits)

Which of the following sets are also subspaces of R3? 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, z) ∈ R3 : x ≥ 0, y ≥ 0, z ≥ 0}

(b) B = {(x, y, z) ∈ R3 : x + y + z = 0}.

(c) C = {(x, y, z) ∈ R3 : x = 0 or y = 0 or z = 0}

(d) D = The set of all solutions to the matrix equation Ax = b, for some matrix A ∈ R3×3 and some

vector b ∈ R3. (Hint: Your answer may depend on A and b.)

Exercise 4 Linear Independence (5+10+15+5 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 .

(d) Let X be another vector space, and let S : W → X be a linear transformation. Define L : V → X as

L(v) = S(T (v)). Prove that L is also a linear transformation.

Exercise 5 Inner Products (5+10 credits)

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

(b) Define 〈·, ·〉 for all x = [x1, x2]T ∈ R2 and y = [y1, y2]T ∈ R2 as

〈x,y〉 = x1y1 + x2y2 ? (x1 + x2 + y1 + y2)

Which of the three inner product axioms does 〈·, ·〉 satisfy?

Exercise 6 Orthogonality (15+6+4 credits)

Let V denote a vector space together with an inner product 〈·, ·〉 : 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.

(c) How do the above statements change if we remove the restriction that x and y have to be

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