Problem #7: suppose that vectors in R3 are denoted by 1 x 3 matrices and define...
16. Let x and y be vectors in R3 and define the skew- symmetric matrix A, by 10-X3 X2 A = X3 0 -X1 I-X2 x 0 (a) Show that x x y = Axy. (b) Show that y x x = Amy.
5) (20 points) a) Show that the vectors x1 = (1, 1, 0)T , x2 = (1, 0, 1)T , x3 = (1, 0, 0)T are linearly independent. Do they form a basis of R3 ? Explain. b) Find an orthonormal basis of R3 using x1 = (1, 1, 0)T , x2 = (1, 0, 1)T and x3 = (1, 0, 0)T .
4. For this question, we define the following matrices: 1-2 0 To 61 C= 0 -1 2 , D= 3 1 . [3 24 L-2 -1] (a) For each of the following, state whether or not the expression can be evaluated. If it can be, evaluate it. If it cannot be, explain why. i. B? +D ii. AD iii. C + DB iv. CT-C (b) Find three distinct vectors X1, X2, X3 such that Bx; = 0 for i =...
3. Suppose that X1, X2, X3 be i.i.d. random variables with P(Xi 0) 2/5 and P(X 1) 3/5. Find the MGFof X, + X2 + X 3. 3. Suppose that X1, X2, X3 be i.i.d. random variables with P(Xi 0) 2/5 and P(X 1) 3/5. Find the MGFof X, + X2 + X 3.
Problem 8. Define a transformation T : R2 + R3 by T(x1, x2) = (–2x1 – 8x2,6x1 + x2, 4x1 – 7x2). (a) Find the standard matrix of T. (b) Find the image of u= under T. - 2 [1] 1 (c) If possible, find a vector x whose image under T is b = [ ། 2 -1
7. In each part of this problem a set of n vectors denoted V, , denoted V. Carefully follow these directions V, is given in a vector space i) Determine whether or not the n vectors are linearly independent. i) Determine whether or not the n vectors are a spanning set of V Then find a basis and the dimension of the subspace of V which is spanned by these n vectors. (This subspace may be V itself.) a. V...
Question 1: Let T: R3 ---> R2 defined by T(x1,x2,x3) = (x1 + 2x2, 2x1 - x2). Show that T as defined above is a Liner Transformation. Question 2: Determine whether the given set of vectors is a basis for S = {(1,2,1) , (3,-1,2),(1,1,-1)} R3 Need answers to both questions.
Check if the given vectors are optimal solutions of the corresponding problems. 1. XI + 4X2 + X3 → max. 4X1 + 11X2 + 3X3 7, rI X2-X3=0, X1+ 13, .y20, j = 1.2.3. Check if the given vectors are optimal solutions of the corresponding problems. 1. XI + 4X2 + X3 → max. 4X1 + 11X2 + 3X3 7, rI X2-X3=0, X1+ 13, .y20, j = 1.2.3.
Consider the following linear transformation T: RS → R3 where T(X1, X2, X3, Xa, Xs) = (x1-x3+Xa, 2x1+x2-x3+2x4, -2X2+3x3-3x4+xs) (a) Determine the standard matrix representation A of T(x). (b) Find a basis for the kernel of T(x). (c) Find a basis for the range of T(x). (d) Is T(x) one-to-one? Is T(x) onto? Explain. (e) Is T(x) invertible? Explain
Consider the following linear transformation T: R5 → R3 where T(X1, X2, X3, X4, X5) = (*1-X3+X4, 2X1+X2-X3+2x4, -2X1+3X3-3x4+x5) (a) Determine the standard matrix representation A of T(x). (b) Find a basis for the kernel of T(x). (c) Find a basis for the range of T(x). (d) Is T(x) one-to-one? Is T(x) onto? Explain. (e) Is T(x) invertible? Explain