Problems
For problem 9-15, determine
(a) Computing
(b) Direct calculation.
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Differential Equations and Linear Algebra (4th Edition)
- 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.arrow_forward1-The set B={3−2x^2, 6−3x−4x^2, 6x−20+14x^2} is a basis for P2. Find the coordinates of p(x)=45−15x−32x^2 prelative to this basis: [p(x)]B= 2- Let f:ℝ2→ℝf:R2→R be defined by f(⟨x,y⟩)=6x+5yf(⟨x,y⟩)=6x+5y. Is ff a linear transformation? a. f(⟨x1,y1⟩+⟨x2,y2⟩)=? (Enter x1x1 as ??x1, etc.) f(⟨x1,y1⟩)+f(⟨x2,y2⟩)= ?Does f(⟨x1,y1⟩+⟨x2,y2⟩)=f(⟨x1,y1⟩)+f(⟨x2,y2⟩) for all ⟨x1,y1⟩,⟨x2,y2⟩∈ℝ^2? b- f(c⟨x,y⟩)= c(f(⟨x,y⟩))= Does f(c⟨x,y⟩)=c(f(⟨x,y⟩)) for all c∈ℝ and all ⟨x,y⟩∈ℝ2 c-Is f a linear transformation?arrow_forwardSuppose T is the transformation from ℝ2 to ℝ2 that results from a reflection over the y-axis followed by a x-shear of 1. Find the matrix A that induces T. A=?arrow_forward
- Suppose T is a transformation from ℝ2 to ℝ2. Find the matrix A that induces T if T is rotation by 5/4π. A = ?arrow_forwardIf {~v1,··· ,~vr} is linearly independent and T is a one to one linear transformation, show that {T~v1,··· ,T~vr} is also linearly independent. Give an example which shows that if T is only linear, it can happen that, although {~v1,··· ,~vr} is linearly independent, {T~v1,··· ,T~vr} is not. In fact, show that it can happen that each of the T~vj equals 0.arrow_forwardSolve for the laplace transformation of y" - 6y' + 9y = t ; y(0) =0, y'(0) = 1arrow_forward
- Suppose T : P2 → R^2 is a linear transformation. If B = {1, x, x^2} andD = {(1, 1),(0, 1)}, find the action of T given MDB(T) = 1 2 −1 −1 0 1 .arrow_forwardGiven the matrix X = 1 0 0 -1 0 3 -1 -2 0 2 0 2 1 0 -1 and consider the linear transformation Vx : R5 -> R3with the given Vx (u) = Xu, for u ∈ R5a.) Shown below is a vector. Is it in the range of Vx? Explain. 1 0 2arrow_forward4 If T : R2 → R2is a linear transformation which has T(e1) = T(e2), then AT haseigenvalue 0.arrow_forward
- Find the matrix [ T] C<--B of the linear transformation T : V ---> W with respect to the bases B and C of V and W, respectively. Verify by computing T(v) directly T: P1--->P1 defined by T ( a + bx) = b -ax, B= {l + x, 1 -x}, C = {l, x}, v = p (x) = 4 + 2xarrow_forwardLet L : P1 → P1 be a linear transformation defined by L(t − 1) = t + 2 and L(t + 1) = 2t + 1. (a) What is L(5t + 1)? (b) What is L(at + b)?arrow_forwardIn this problem, allow T1: ℝ2→ℝ2 and T2: ℝ2→ℝ2 be linear transformations. Find Ker(T1), Ker(T2), Ker(T3) of the respective matrices:arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning