Let k be an integer such that the vectors and 3 are linearly dependent. What must...
8. (10 Pts) Answer by True / False and justify your answer. (a) Let A be a 2 × 2 matrix such that(A2-Nthen, if A ±1 A--. (b) If C is a skew-symmetric matrix of odd order n, then |C-0 (c) If A is a square matrix, and the linear transformation L(z) Az is one-to-one, then the linear transformation x ? At is also one-toone. z), ? O (z, y, z) = (az, ay, 0), then V is not a...
Prove: Let k be a positive integer, and set n :=2k-1(2k – 1). Then (2k+1 – 1)2 = 8n +1 Prove: Let n be a positive integer, and let s and t be integers. Show that Hire (st) = n(s) in (t) mod n.
Let m be a positive integer. Show that a mod m - b mod m t a - b (mod m) Drag the necessary statements and drop them into the appropriate blank to build your proof (mod m Dag the mecesary eemnes a ohem int the approprite Proof method: Proof assumptions), at-qm + Proof by contradiction aaandh mam it Implication(s) and deduction(s) resulting from the assumption(s): a mk + bmk Hqm tr a-(k + q)m+ r Conclusion(s) from implications and...
Let A = 1 -2 31 2 1 -k . Find the value of k for which det A = 10. Your value of k should be an integer. Lo –3 kJ Answer:
Let T' be a linear transformation represented by Г-3-51 T3-1 2 -3 Check ALL vectors which are in the range of T. O(2, -1) O(33, 3, 20) 0 3, - 4, 3 O(6, -4, -3)
Q 3 a) Let n > 2 be an integer. Prove that the set {z ET:z” = 1} is a subgroup of (T, *). Show that it is isomorphic to (Zn, + mod n). b) Show that Z2 x Z2 is not isomorphic to Z4. c) Show that Z2 x Z3 is isomorphic to 26.
11 -2 31 Let A = 2 1 k Find the value of k for which det A = -10. Your value of k should be an integer 10 -3 -k] Answer: Check Let [ 2 2 -1] A = -1 -11 ( 2 4 -1] Given that 1 is an eigenvalue of A, find a value of k so that (5.-1,k) is in the eigenspace of A corresponding to the eigenvalue I.
[1 41 and we [-121 (1 point) Let A= 3 12 Find k so that there exists a vector x whose image under the linear transformation T(x) = Axis w. Note: The image is what comes out of the transformation. k= Find k so that w is a solution of the equation Ax = 0
Q4 Let z = dkdk-1 d2dı be the base 10 representation of an integer x where di,..., dk are digits drawn from 0,...,9. Explain why x d1 + d2 + . . . + dk (mod 9) = so, also, z di + d2 + . . . + dk (mod 3) = and Thus for example to check whether 57,711 is divisible by 9 or 3 we just add up the digits 5 + 7+7+ 1 + 1 =...
nsid r the following et ār vnctors. Let 1 v2 and V3 be column vectors in and let A be the 3 × 3 matrix v 1 v2 v③ with these vectors as its columns. The vi v2 and ] are linearly dependent if and nly the hom 9ene us linear system with augmented matrix 시 has a no tr ia solution Consider the following equation. 81-3-311 Solve for ci 2, andc3. If a nontrlvial solution exists, state it or...