It follows that M1 is invariant subspace of operator ProjM2. Indeed, if v∈M1 then
ProjM1ProjM2v=ProjM2ProjM1v=ProjM2v,
so
ProjM1(ProjM2v)=ProjM2v,
but this can happen only if ProjM2v∈M1.
Similarly, it can be proved that M⊥1 is invariant for ProjM2, and also M2,M⊥2 are invariant for ProjM1 by symmetry.
Since M1 is invariant for ProjM2 and ProjM2 has two eigenspaces M2,M⊥2, then M1 can be split into
M1=(M1∩M2)⊕(M1∩M⊥2)
Similarly,
M2=(M2∩M1)⊕(M2∩M⊥1)
Now, clearly, (M1∩M⊥2)⊥(M2∩M⊥1), so
M1+M2=(M1∩M2)⊕(M1∩M⊥2)⊕(M2∩M⊥1),
hence
Pr(M1+M2)=Pr(M1∩M2)+Pr(M1∩M⊥2)+Pr(M2∩M⊥1)=
=Pr(M1)+Pr(M2)−Pr(M1∩M2)
5. Let CONTAINPDA DFA L(M1) C (M2)}. Show that CONTAIN PDA DFA is decidable. {{M1, M2) M1 is a PDA and M2 is a DFA such that = 5. Let CONTAINPDA DFA L(M1) C (M2)}. Show that CONTAIN PDA DFA is decidable. {{M1, M2) M1 is a PDA and M2 is a DFA such that =
1. State whether the following are part of M1 but not M2, M2 but not M1, part of both M1 and M2, or are not part of either M1 or M2: a. $600 in your checking account (2) b. $1,000 line of credit with your credit card (2) c. $2 million held in a commercial bank (2) d. $1000 held in a money market fund online (2) e. Three dollars and four nickels in your pocket (2)
2.(a) Show that if 7 1 0 0 M 00). and M3 = 10 1 = Mi= 100) M2 (01) then the span of {M1, M2, M3} is the set of all symmetric 2 x 2 matrices. (b) For k = 0,1..., n let px(x) = zk + 2k+1 + ... + x Show that the set {Po(2), p1(2),..., Pn(x)} is linearly independent in Pn(F).
Let M1, M2 and M3 are linearly independent sets of vectors, and that M, C M, CM2. Let M = M, UM, U M3. Then, which of the following is always true? a) M is linearly dependent b) M is linearly independent c) M is both linearly dependent and linearly independent d) M is neither linearly dependent nor linearly independent
Consider three masses, m0, m1, and m2 located at (x,0), (0,y), and (0, -y) respectively: a.) Find the total gravitational force on m0 due to the other two masses. b.) Supoose that m1 = m2 and that y>>x, if m0 were allowed to move, show that the motion would be osillatory about the origin. What would the frequency of oscillation be?
The null hypothesis for the independent-measures t-test states _____ A) M1 - M2 = 0 B) μ1 - μ2 ≠ 0 C) μ1 - μ2 = 0 D) M1 - M2 ≠ 0
Two blocks of mass m1 and m2 > m1 are drawn above. The block m1 sits on a frictionless inclined plane tipped at an angle θ with the horizontal as shown. Block m2 is connected to mı by a massless unstretchable string that runs over a massless, frictionless pulley to hang over a considerable drop. At time t = 0 the system is released from rest. a) Draw a force/free body diagram for the two masses. b) Find the magnitude of the...
2. Let w-a b :a 2b-7c, a subset of M2 2x2 c d a) (5 points) Find a set of "vectors" in M2-2whose span is W b) (5 points) Show that the vectors in part (a) are linearly independent. c) ( point) Since Wis the span of the vectors obtained in part (a), we can conclude that W isa of M2x2. (See Theorem 4.1.2 in the 4.1 Part 2 notes)
o 1 0 -1 Exercise 2. Let A= in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R3, set g(W) = WT AT ER. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]C,B in the - 1 bases C = {1} and B { 8.00 } ? (ii) Let f : R3 → R be the function defined by f() = 7T Aw E R. Show that...
Problem 2 120 pointsl: For the multiplexing system shown in the figure M1(o) MiC) -5000 M2(0) m20) 2 cos 20,000 0 5000 0) 2 cos 10,000 Sketch signal spectra at points a, b, and c. a. b. What ust be the bandwidth of the channel in rad/sec? (i.e. what is the total bandwidth at point c. c. Design a coherent receiver to recover signals m() and m() from the modulated signal at point (In other words, sketch the block diagram...