这是一篇来自澳洲的关于机器学习入门的机器学习代写案例分享
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 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 h·, ·i is symmetric and linear in the fifirst argument, then it is bilinear.
(b) Defifine h·, ·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 h·, ·i satisfy?
Exercise 6 Orthogonality
(15+6+4 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.
(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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