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- Solve 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_forwardPlease, help with this problem.arrow_forwardGive the properties for the equation x 2 - 3y 2 - 8x + 12y + 16 = 0 Center (4, 2) (4, -4) (4, -2)arrow_forward
- 1. Find all solutions: 2. In what ways can 88-8 and be written x+y ==0 a as a linear combination of [] [] [4], and ² 3. Show that the vector 2 is NOT contained in the span of the following three vectors 6 4. For which b are the vectors -0.0 x+2y+z=5 2x+3y-z=3 independent?arrow_forwardProblem 5. Let a E R" be fixed. Suppose that vectors x, y E R" are related by the equation а — х+ (x:у)у. (a) Show that ||a||² – ||x||² 2 + ||y||² - (x - y)? (b) Deduce that ||a|| > ||x||.arrow_forwardFormulate this problem as an LP (do not solve, just formulate please) There are currently 3 machines in a factory: P1, P2, and P3. Assume that this rectilinearshaped factory is located on the first quadrant of the coordinate system and onecorner is at the origin (point (0,0)). The coordinates of the existing machines are asfollows:P1=(10,15), P2=(20,25) ve P3=(40,5)To meet the increasing demand and respond to changing customer demands, thecompany decided to grow and acquired two new machines: N1 and N2. When thesetwo machines are put into operation, they will exchange materials with each other andwith the other three machines. 400 units of material will be transported between the 2two new machines in a week. Similarly, 400 units will be transported between N1 andP1, 0 between N1 and P2, and 500 units between N1 and P3. The transportationbetween N2 and P1, P2, and P3 are 200, 100, and 0, respectively.Materials are transported with an overhead crane. This crane can move linearly in xand…arrow_forward
- Find the point on the line y = 2x + 3 that is closest to theorigin.arrow_forward1. Describe the solutions of the following system in parametric vector form. 2x1 + 4x2 – 6x3 + x4 x1 – x2 + 4x3 + x4 = 0 -x1 + x2 – x3 + x4arrow_forwardThere are three base stations (M1, M2 and M3) that can send and receive signals from your mobile phone (at location Z). Suppose that in a cartesian coordinate system these base stations are located at (0,0), (36, 0) and (16,32), respectively, find the coordinates of Z. The distances between you (at point Z) and the base stations are 29 km, 25 km and 13 km, respectively. (Note: 1 map unit is equivalent to 1 km distance)Hint: You may use graphical method by plotting the points or using this mathematical equation: (?−?′)?+(?−?′)?=??Where: x and y = coordinates of unknown pointx’ and y’= coordinates of known pointr = distance between unknown and known point PLEASE GRAPH AND SHOW POINTSarrow_forward
- Suppose that A= 2 6 2 [-1 1 1] Describe the solution space to the equation Ax = 0. Describe the solution space to the equation Ax = b where b : Are there any vectors b for which the equation Ax = b is inconsistent? Explain your answer. Do the columns of A span R? Explain your answer.arrow_forwardi need help on parts a-g.arrow_forward1. A line passes through the points A (7,-4) & B (-5, -2). Find a vector equation of the line. [x.y] = [7,-4] + t[−5,−2] b) [x.y] = [7,-4] + t[-2,-6] c) [x.y] = [-5, -2] + t[-12,2] d) [x.y] = [12,-2] + t[7,4] 2. A line has slope-3 and x-intercept 5. Find a vector equation of the line. a) [x.y] = [-3, 1] + t[5,0] b) [x.y] = [0,5] + t[−3,1] c) [x.y] = [5,0] + t[-1,3] d) [x.y] = [-1, -3] + t[0,5] 3. Write the scalar equation of the plane with normal vector n = [3,2,1] and passing through the point (4, 5, 6). 28 = 0 a 3x + 2y + z b) 4x + 5y + 6z+ 28 = 0 c) 3x +2y+z+28 = 0 d) 4x + 5y + 6z - 28 = 0 4. A plane passes through the origin and has the direction vectors [-1, -2, -3] and [-1, 3, -2]. Find a scalar equation of the plane. a) -x - 2y3z = 0 b) 13x + y - 5z = 0 c) -x + 3y2z = 0 d) 5y + z = 0arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage