1. Let az, az, az, a4 are vectors in R3. Suppose that az 3a1 – 2a3 + 84. (a) Are aj, aj, az, a4 linearly independent? (b) Suppose that ai, az, a4 are linearly independent. What is the dimension of the span{a1, az, az, a4}? (c) Is the set of vectors aj, az, az, a4 form a basis of R3? Explain your reasoning. (d) Form a basis of R3 using a subset of ai, a2, a3, 24.
13. Consider the sequence of numbers ao, ai, a2, a3, given by ao-2, ai-3, and for any positive integer k 2, a3ak 2ak-1. (a) Evaluate a2,a3, a4,as. Show your work. (b) Prove that for all positive integers n, an 2 +1
1) Consider the switching networks shown in Fig. 1. Let Ai, A2, and As denote the events that the switches s1, s2, and s3 are closed, respectively. Let Aab denote the event that there is a closed path between terminals a and b. Express Aab in terms of Ai, A2, and A3 for each of the networks shown. 2 Figure 1
3. Consider the graph with 8 nodes A1, A2, A3, A4, H, T, F1, F2. Ai is connected to Ai+1 for all i, each Ai is connected to H, H is connected to T, and T is connectedto each Fi, Find a 3-coloring of this graph by hand using the following strategy: backtracking with conflict-directed backjumping, the variable order A1, H, A4, F1,A2, F2, A3, T, and the value order R, G, B.
2.1.2. Let A = {(x ,y): x 2, y s 4}, A2={(x,y): x2, y < 1}, A3={(x,y): x <0, y <4}, and A4={(x,y): x 0, y < 1} be subsets of the space A of two random variables X and Y, which is the entire two- dimensional plane. If P(A) 7/8, P(A2) = 4/8, P(A3) =3/8, and P(A4)= 2/8, find P(As), where As={(x, y) :0 <x <2, 1< ys 4}.
Matrix notation: A=(a1,a2,a3.....,an) = [a1 a2 a3 a4 .....an] are they equal? look at the sample picture A should be matrix but it uses ( ) rather than [ ] Given that A is an n×n matrix with the property AX = 0 for all X " 1 A=(a,,a,, 0 0 Let a.) Let e, =| | | ← ith element Comment
Let {dn}n≥0 denote the number of integer solutions a1 +a2 +a3 +a4 = n where 0 ≤ ai ≤ 5 for each i = 1, 2, 3, 4. Write the ordinary generating function for {cn}n≥0. Please express the ordinary generating function as a rational function p(x) /q(x) where both p(x) and q(x) are polynomials in the variable x.
Let V be the vector space of all sequences over R. Given (a1, a2, T,U V V by ) e V, define : ) ...) = (0, a1, 0, a2, 0, a3, . . . ) Тај, а2, аз, ад, 0, аз, (a1, a3, a5,.) and U(a1, a2, a3, a4, (a) Find N(T) and N(U) (b) Explain why T is onto, but not 1-1 (c) Explain why U is 1-1, but not onto.
Need help with this ASAP 10 Q3. (a) Prove the Bonferroni Inequality on three events Ai, A2 and A: P(AinAnAS) 21- P(A) - P(A2)- P(As) (b) Using the results in Q3.(a), and clearly describing the events Ai, A2 and A3, construct a 100(1-a)% joint confidence intervals for estima- tion of three parameters, denoted by 0,02 and 03, say. 10 Q3. (a) Prove the Bonferroni Inequality on three events Ai, A2 and A: P(AinAnAS) 21- P(A) - P(A2)- P(As) (b) Using...