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COMP3670/6670: Introduction to Machine Learning
创建日期：2024-07-29 17:38:16
标签： COMP3670
Introduction to Machine Learning
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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.
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.
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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