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- Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if the matrix [x1y11x2y21x3y31] has rank less than 3.arrow_forwardFind a basis for R2 that includes the vector (2,2).arrow_forwardIn a standard Cartesian coordinate system a vector, R, has an x-component of + 5.0 m and a y- component of - 4.0 meters. In the same coordinate system a vector F has an x-component of - 3.0 N and a y-component of 2.0 N. The cross product RxF is equal to 2.0 m*N in the + z direction 2.3 m*N in the - x direction 2.0 m*N in the - z direction 2.8 m*N in the + x direction 23 m*N in the - y directionarrow_forward
- Problem 6. Suppose that V₁, V2 and v3 are linearly independent vectors in a vector space V. Prove that the vectors W₁ = V₁ + V2, W2 = V₂ + V3 and w3 = V3 + V₁ are also linearly independent in V.arrow_forward1. Write the vector H- 16 as a linear combination of the vectors and Darrow_forward3. If a = 3x - 5ỹ and b = -2x +9y, write the vector 5a-86 in terms of x and yarrow_forward
- If T is defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is - 6 1 – 4 2 unique. Let A=| 0 1 -3 and b=| - 17 4 - 17 8 Find a single vector x whose image under T is b. x = Is the vector x found in the previous step unique? A. No, because there are no free variables in the system of equations. B. Yes, because there is a free variable in the system of equations. C. Yes, because there are no free variables in the system of equations. D. No, because there is a free variable in the system of equations.arrow_forwardThe following question is from linear algebra : Factors the vector (6, -5, -1)t into three components a,b,c that satisfy the following conditions: a depends on (2,0,1)t, b depends on (1,2, 0)t and c is orthogonal to a and b. Please show it step by step.arrow_forwardVII. For R3 , find the Change of Basis matrix from basis B to basis D. As a mild form of verification, check your result for the vector i = 10 Show all work neatly, in an organized manner. Show all work, even when finding the representations of vectors relative to particular bases. If your matrix does not produce the correct result, you must find and correct all errors or lose credit for this problem. B = -2 2 D = 2 3arrow_forward
- A B -1 -2 C D E -4 -5 -3 F 1 GHI 2 3 L JK -1 -2 4 5 P Q R 2 3 1 MNO -5 -4 -3 TU S 4 5 -1 V -2 W X -3 -4 Y 1 N 2 Use the table above to create two vectors in 2-Space: * and y Let * be a vector representative of the first two letters in your given name and y represent the first two letters of your surname. (Ex. Albert Einstein: x= [-1, -2] and y = [-5,4]) My vectors are 2 = [-1₁-4] and j = [ -5, -2] Use the table above to create two vectors in 3-Space: è and f Let è be a vector representative of the first three letters in your given name and represent the first three letters of your surname. (Ex. Albert Einstein: € = [-1, -2, -2] and f = [-5,4,-4]) My vectors are è = [-1,-4, 2] and f= [-5, -2₁ -2] e 1. Angle Between Vectors and Projects a) Use the dot product to verify the type of angle connecting the vectors and y (acute, right or obtuse). b) Find the angle connecting the vectors and y in two different ways. c) Use the dot product to verify the type of angle connecting the vectors e and…arrow_forwardLet B = 3 (1, x + 8, (x − 1)², 2x³) be an ordered basis for P3. Find the coordinate vector of 3 - f(x) = 8x³ — 6x² - 7x + 3 relative to B. fB =arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning