Concept explainers
Problems
For problem 9-15, determine
(a) Computing
(b) Direct calculation.
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Differential Equations and Linear Algebra (4th Edition)
- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forwardLet T:P2P4 be the linear transformation T(p)=x2p. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3,x4}.arrow_forwardLet T be a linear transformation from R2 into R2 such that T(1,0)=(1,1) and T(0,1)=(1,1). Find T(1,4) and T(2,1).arrow_forward
- Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.arrow_forwardSuppose T: R³-R² is a linear transformation. Let U, V and W be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(W). -3 3 2 3 ਪ = -3 -1 0 T(W) = 0 0 V = -3 W = | 3 - 1 T(I) = 0 T(V) = 0 -3arrow_forwardSuppose f(en) = a₁ for n = 1,2 and f is a linear transformation. A = = 3 im(f) 2 1 = -1 -2 -3 -3 -2 -1 19 e2 e1 1 2 3 Domain X 3 2 1 -1 -2 -3 a2 -3 -2 -1 4 a. Find a formula for f. Your answer should be a coordinate vector with the variables x and y in its components. f(x, y) a1 1 2 Codomain b. Find the matrix for the linear transformation f (relative to the standard basis in the domain and codomain). That is, find the matrix A such that f(x) = Ax. For [1 2 instance, enter [[1,2], [3,4] ] for the matrix 3 4 3 X c. Find the kernel of f. Enter your answer as a vector with constant entries, a vector with the variables x or y (or both) in its components (using a minimum numbe of variables), or enter R^2 for R². ker(f) = d. Find the image of f. Enter your answer as a vector with constant entries, a vector with the variables x or y (or both) in its components (using a minimum numbe of variables), or enter R^2 for R².arrow_forward
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