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For Problems 15-18, determine whether the given set
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- For each of the following lists of vectors in R3, determine whether the first vector can be expressed as a linear combination of the other two. (a) (-2,0,3) ,(1,3,0),(2,4,-1) (b) (1,2,-3) ,(-3,2,1) ,(2,-1,-1) (c) (3,4,1) ,(1,-2,1), (-2,-1,1) (d) (2,-1,0) , (1,2,-3), (1,-3,2) (e) (5,1,-5) , (1,-2,-3), (-2,3,-4) (f) (-2,2,2) ,(1,2,-1) ,(-3,-3,3)arrow_forwardSolve the following exercises, you will need to show all your work to receive full credit. Consider the matrix, 2 1 -2 2 3 -4 1 1 1 - Knowing that f(t) = (t – 1)²(t - 2) is the characteristic polynomial, do the following: 1. find a basis of eigenvectors; 2. Find P such that P- AP is a diagonal matrix D. Give Darrow_forward4. a. If 37– 2ỷ = đ and 53 – 3ỷ = b, express the vectors i and y in terms of a and b. %3D b. Solve for a, b, and c: (2, –1, c) + (a, b, 1) – 3(2, a, 4) = (-3, 1, 2c). %3Darrow_forward
- 1. Given the vectors à = (-1, 1, 1), b = (2, 1, –3), and ở = (5, 1, –-7), calculate the value of each of the following: a. d xB b. Bx 2 à-(5 x a) (a x 8) × (5 x 2) с. а d.arrow_forwardExpress each of the following vectors in R² as linear combinations of the vectors [3] and [3] (a) (b) (c) (d) 7 13 H 5 5 12 22 = || = = + + + + + 5 r 3 rarrow_forwardLet A = - - 3 [1] [2] and b = 3 9 b2 Show that the equation Ax = b does not have a solution for some choices of b, and describe the set of all b for which Ax=b does have a solution. How can it be shown that the equation Ax=b does not have a solution for some choices of b? A. Find a vector x for which Ax=b is the identity vector. B. Row reduce the augmented matrix [ A b] to demonstrate that A b has a pivot position in every row. C. Find a vector b for which the solution to Ax=b is the identity vector. D. Row reduce the matrix A to demonstrate that A does not have a pivot position in every row. E. Row reduce the matrix A to demonstrate that A has a pivot position in every row.arrow_forward
- Given the following vectors, X.= X.= X,= 1 -3 Determine whether the vectors are independent of each other or not by testing B*,+B,x,+B+B=0, for all ß,. B,. B3, and ß , Question: In continuation from the previous question, express x, in terms of x,, x,, and x, by entering your answers: x2 = X1 + X3 + X4 Special instructions: Enter answers in fraction form only. Express in lowest terms. Other answers will not be accepted. Example: 3, -7, 23/34, -43/11, etc. Not accepted: 3.3333, -1.2345 3. 2.arrow_forwardFor each of the following lists of vectors in R³, determine whether or not the first vector can be expressed as a linear combination of the other two. (a) (-2, 0,3), (1, 3,0), (2, 4, -1) (b) (1,2,3), (-3, 2, 1), (2, -1, −1) (c) (3, 4, 1), (1, -2, 1), (−2, —1, 1)arrow_forwardIf k is a real number, then the vectors (1, k), (k, k+ 56) are linearly independent precisely when k # a, b, where a = , 6 = and a < b.arrow_forward
- Define and x as the vectors y and x= [0.5, 1, 1.5, 2, 2.5] y=[0.8, 1.6, 2.4, 3.2, 4.0]. Then use them in the following expressions to calculate z using element-by-element calculations. (a) z = x² + 2xy 3 4 (b) z=xye*- xy +8.5arrow_forwardIf v + w = [ 5/1 ] and v - w = [ 1/5 ],compute and draw the vectors v and w.arrow_forwardIf k is a real number, then the vectors (1,k),(k,7k+8) are linearly independent precisely when k ≠ a,b, where a = ? and b = ?, and a<barrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage