(4)A basis for R³ consists of the three vector V₁ =--0--0V3V2 =113-12 (You do not have to verify that ß is a basis forR³.)There is a linear transformation T: R³ R³ which satisfies T(v₁) = 2v1,→T(v₂) = -5v3, T(v3) = V₁ -2v2. Compute the determinant of T.=

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Answered: (4) V2 = A basis for R³ consists of the…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 25CM: Find a basis B for R3 such that the matrix for the linear transformation T:R3R3,...

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Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
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Do the vectors v1 = (-3, 7) and v2 = (5, 5) form a basis for R²?
Q3. a) Express the vector (4,2) eR' as linear combination of the vectors (1,1) and (1,-1). b) Let T be a linear transformation from R? to R²such that T(1,1)=(-1,2) and T (1,–1)=(0,4). Find T (4,2). c) Find (3u– 2v) (u-3v) given that Ju|=3, u-v=7 and ||v|| = 4 . %3D
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7. (a) Show that the vectors v1 = (0, 3, 1, – 1), v2 = (6, 0, 5, 1), and v3= (4, – 7, 1, 3) form a linearly dependent set in R4 (b) Express each vector as a linear combination of the other two.
2- We can write the vector x = (19,–1, -) in R³ as a linear combination of the vectors x1 = (4,2, –1), x2 = (1,5,–1) and x3 = (2,-2,3), and the scalars are A:a, = -,a, = -,a, =-2 -,a2 = 5, az C: az = 5, az = -2,az B: a, = = 2 1 2 A В C
Suppose T: R¹→P3 is a linear transformation whose action on a basis for R4 is as follows: -3 a C = II 9x²-12x+6 T 0 0 Describe the action of T on a general vector, using x as the variable for the polynomial and a, b, c, and d as constants. Use the '^' character to indicate an exponent, e.g. ax^2-bx+c. -4x²+4x-4 T −4x³−8x²+14x−14 T 2x³-6x²+5x-2
Let 7 = A = The cross product of two vectors in IR³ is defined by -H . Find the matrix A of the linear transformation from R³ to R³ given by T(Z) = x 7. [α₂b² - αzb₂] [-]-[6] xb₂ = ab₁-a₂b₂ b₂ [a₁b₂ − a₂b₁]

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(4) V2 = A basis for R³ consists of the three vector V₁ = , V3 = 1 2 3 " (You do not have to verify that is a basis for R³.) There is a linear transformation T : R³ → R³ which satisfies T(v₁) = 2v₁, T(v₂) = −5v3, T(V3) = V1 — 2Vv2. Compute the determinant of T.