COT 5615 Fall 2020
Midterm I
(All material inside the Canvas page for the class is permitted)
1. (30 points) You are given two sets of vector space bases {xi}K1i=1 with xi ∈RK1 and
{
y j
}K2
j=1 with y j ∈RK2 .
From these you construct a new set of vectors with components zi j,ab ≡ xiay jb for a ∈ {1, . . . ,K1} and
b ∈ {1, . . . ,K2}. In other words, you multiply the ath component of xi and the bth component of y j to form
the abth component of a vector zi j in RK1K2 .
• (15 points) Is the new set of K1K2 vectors a basis in RK1K2? Justify your answer showing all steps.
• (10 points) If in addition to the knowledge that {xi}K1i=1 with xi ∈ RK1 and
{
y j
}K2
j=1 with y j ∈ RK2 are
bases, you are given that both sets of vectors are orthogonal bases, is the new set
{
zi j
}
orthogonal?
Justify your answer showing all steps.
• (5 points) If in addition to the knowledge that {xi}K1i=1 with xi ∈ RK1 and
{
y j
}K2
j=1 with y j ∈ RK2 are
bases, you are given that both sets are orthonormal bases, is the new set
{
zi j
}
orthonormal? Justify
your answer showing all steps. (You do not need to repeat the orthogonality aspect in the answer to
this question.)
2. (30 points) Show that ‖A‖1 and ‖A‖∞ satisfy the following properties.
• ‖A‖> 0 if A 6= 0 (the all zero matrix).
• ‖γA‖= |γ| · ‖A‖ for any scalar γ .
• ‖A+B‖ ≤ ‖A‖+‖B‖.
• ‖AB‖ ≤ ‖A‖ · ‖B‖.
• ‖Ax‖ ≤ ‖A‖ · ‖x‖ for any vector x (and for the same choice of vector norm).
3. (25 points) Assume an m×nmatrix A of rank n (and m> n) and let the reduced SVD of A=UΣV T with
U being m×n, Σ being n×n and V being n×n.
• (5 points) Write P= A(ATA)−1AT in terms of the reduced singular value decomposition matrixU .
• (5 points) Interpret your result taking care to explain the relationship between your expression for P
and the rank of the matrix.
• (15 points) Then rewrite the least-squares orthogonality principle for the problem minx ‖b−Ax‖22 in
the form Cx = Db where C and D are written in terms of the SVD of A. C should not contain U and
D should not contain V . Show all steps and interpret your result in terms of the two vector spaces Rm
and Rn.
4. (15 points) You are given a set T = {T1,T2, . . . ,TN} of numbers. Given T , you compute a scalar quantity
w≡ 1β log ∑Nk=1 exp{βTk} and a scalar quantity z≡ ∑Nk=1Tk exp{βTk}∑Ni=1 exp{βTi} where β > 0.
• Derive a mathematical relationship between w and z. Give a detailed explanation. No assumptions
regarding the number of ties in T need be made since these can be handled using special cases.
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