(a). B is not a basis for R, the set of real numbers as a singleton matrix is not the same as a number.
(b). The set B is linearly dependent as (1,1)T = (1,0)T+(0,1)T. Hence B is not a basis for R2.
( c). Since dim(R3 ) = 3, and since B contains only 2 vectors, hence B is not a basis for R3.
(d). Let A be the matrix with the vectors in the set B as columns. Then A =
1 |
3 |
0 |
1 |
4 |
5 |
0 |
2 |
10 |
The RREF of A is
1 |
0 |
-15 |
0 |
1 |
5 |
0 |
0 |
0 |
This implies that (0,5,10)T= -15(1,1,0)T+5(3,4,2)T. Thus, the set B is linearly dependent. Therefore, B is not a basis for R3.
(e). The set B is linearly independent and the vectors in B spans R2. Therefore, B is a basis for R2.
(f). Let A be the matrix with the vectors in the set B as columns. Then A =
1 |
0 |
0 |
0 |
=1 |
1 |
0 |
0 |
0 |
-1 |
1 |
0 |
0 |
0 |
-1 |
1 |
The RREF of A is I4 which implies that the set B is linearly independent and the vectors in B spans R4. Therefore, B is a basis for R4.
In each of the cases below, is the set B a basis of R"? Explain why or why not. (a) {[2]} (b) n = 1 and B n = 2 and...
2, For each of these sets. A={3n : n E N), B = {r E R : x2 < 7), and C = {x E R : x < 12), (i) Is the set bounded above? Prove your answer.] ( .] ii) Is the set bounded below? Prove your answer answer the following questions:
determine if each set of quantum numbers below is permissible or not. if yes, write the orbital designation for each, if not explain why. 4. Determine if each set of quantum numbers below is permissible or not. If yes, write the orbital designation for each, if not explain why. a) n=2 1=1 ml= +1 b) n=11=0 ml=-1 c) n=4 = 2 ml=+1 d) n=31= 3 ml=0
s={(8.60) :) :) is a basis of M3x2(R)? (d) (1 point) The set = {(1 9:(. :) : 6 1) (1 1) (1 :) :()} is linearly independent. (e) (1 point) For a linear transformation A:R" + Rd the dimension of the nullspace is larger than d. (f) (1 points) Let AC M4x4 be a diagonal matrix. A is similar to a matrix A which has eigenvalues 1,2,3 with algebraic multiplicities 1,2, 1 and geometric multiplicities 1,1, 1 respectively. 8....
If x is a binomial random variable, compute p(x) for each of the cases below. a. n=4, x=1, p=0.4 b. n=6, x=3, q=0.6 c. n=3, x=0, p=0.8 d. n=4, x=2, p=0.7 e. n=6, x=3, q=0.4 f. n=3, x=1, p=0.9
Hi, could you post solutions to the following questions. Thanks. 2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
If x is a binomial random variable, compute p(x) for each of the cases below. a. n=5, x=2, p=0.3 b. n=6, x=3, q=0.2 c. n=4, x=1, p=0.7 d. n=5, x=0, p=0.4 e. n=6, x=3, q=0.8 f. n=4, x=2, p=0.6
(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u, v e R". Show that ul if and only if ||ü + 해2 (c) Let W be a subspace of R" with an orthogonal basis {w1, ..., w,} and let {ö1, ..., ūg} 22 orthogonal basis for W- (i) Explain why{w1, ..., üp, T1, .., T,} is an (ii Explain why the set in (i) spans R". (iii Show that dim(W) + dim(W1) be...
linear algebra 2 part mcq part a part b r(A) Find and n(A) A = 1 - 3 4 -1 9 -2 6 -6 -1 -10 -39 -6 -6 -3 3 -94 9 0 a. r(A) = 5 n(A) = 0 b. r(A) = 3 n(A) = 2 c. (A) = 0 n(A) = 5 d. r(A) = 1 n(A) = 4 e. r(A) = 4 n(A) = 1 f. r(A) = 2 n(A) = 3 А Diagonalize A =...
Question 2 For each of the following relations R, determine (and explain) whether R is: (1) reflexive (2) symmetric (3) antisymmetric (4) transitive (a) R-(x, y):x +2y 3), defined on the set A 10, 1,2,3) (b) R-I(x, y): xy 4), defined on the set A (0,1,2,3,4 (c) R-(x, y): xy 4), defined on the set A-0,,2,3) Question 2 For each of the following relations R, determine (and explain) whether R is: (1) reflexive (2) symmetric (3) antisymmetric (4) transitive (a)...
Q3: Given a relational schema R = {A,B,C,D,E,F,G,H,1,J,K} and a set of functional dependencies F {A B C D E, E F G H I J,AI →K} and a key(R) = AI = 1. Is R in BCNF? If yes, justify your answer [5 points] 2. If no, explain why and decompose R for two levels only [10 points] 3. Check whether the decomposition in step 2 dependency preserved or not [5 points]