# 机器学习代写｜COMP3670/6670: Introduction to Machine Learning

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

(a)A = [2 7 1

1 4 3

0 2 5]

, b = [112](b)A = [1 2 2

3 4 3

], b = [ 105 ]

Exercise 2 Inverses

(4 credits)

Find the inverse of the following matrix, if an inverse exists.

[1 1 2

2 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 satisfified, 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 coeffiffifficients

{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 .

Defifine 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. Defifine 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 , ·i is symmetric and linear in the fifirst argument, then it is bilinear.

(b) Defifine , ·i for all x = [x1, x2]T R2 and y = [y1, y2]T R2 as

h x, yi = x1y1 + x2y2 (x1 + x2 + y1 + y2)

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

Exercise 6 Orthogonality

(15+6+4 credits)

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

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

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