Let V = R4 and let T : V → V be defined by T x1 x2 x3 x4 = x1 −x3 x2 + x4 x1 −x3 −x2 −x4 . (a) Show that T is a linear transformation. (b) Show that T(T(v)) = 0 for v ∈ V . (c) Show that imT = hT(e1), T(e2)i. (d) Show that kerT = imT.
Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1 Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1
; Let at be a linear transformation as follows : T{x1,x2,x3,x4,x5} = {{x1-x3+2x2x5},{x2-x3+2x5},{x1+x2-2x3+x4+2x5},{2x2-2x3+x4+2x5}] a.) find the standard matrix representation A of T b.) find the basis of Col(A) c.) find a basis of Null(A) d.) is T 1-1? Is T onto?
X1 Let x = V = and v2 - and let T: R2R2 be a linear transformation that maps x into xxv, + XxV2. Find a matrix A such that T(x) is Ax for each x. X2 A= Assume that is a linear transformation. Find the standard matrix of T. T:R3-R2, T(41) = (1,3), and T(62) =(-4,6), and T(03) = (3. – 2), where e1, 22, and ez are the columns of the 3*3 identity matrix. A= (Type an integer...
Let T R3 R4 be the linear transformation defined by T(π1, Ο2, 73) - ( 3α1 -4 , X3, 12.x2 3.x3, 6x1-25x3, 10x2 + 10x3) (a) Determine the standard matrix representation of T (b) Find a basis for the image of T, Im(T), and determine dim(Im(T)) (c) Find a basis for the kernel of T, ker(T), and determine dim(ker(T))
Please show work Consider the following linear transformation T: RS → R3 where T(X1, X2, X3, X4, Xs) = (x1-X3+X4, 2x1+x2-X3+2x4, -2x1+3x3-3x4+xs) (a) Determine the standard matrix representation A of T(x). (b) Find a basis for the kernel of T(x). (c) Find a basis for the range of T(x). (d) Is T(x) one-to-one? Is T(x) onto? Explain. (e) Is T(x) invertible? Explain
Consider the subspace W C R4 given by X1 X2 W = ER4 X1 + x2 + x4 = 0 and x2 + x3 + x4 = 0 X3 X4 = Find an orthonormal basis H {h1, h2, h3, h4} for R4 with the property that hy and h2 are elements of an orthonormal basis for W, where orthonormality is defined with respect to the dot product on R4 x R4.
Consider the subspace W C R4 given by X1 X2 W = ER4 X1 + x2 + x4 = 0 and x2 + x3 + x4 = 0 X3 X4 = Find an orthonormal basis H {h1, h2, h3, h4} for R4 with the property that hy and h2 are elements of an orthonormal basis for W, where orthonormality is defined with respect to the dot product on R4 x R4.
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
Consider the following linear transformation T: RS → R3 where T(X1, X2, X3, X4, X5) = (x1-X3+X4, 2X1+X2-X3+2x4, -2X1+3x3-3x4+x5) (a) Determine the standard matrix representation A of T(x).
linear algebra Let T: P2 - P4 be the linear transformation T() = 2x2p. Find the matrix A for T relative to the bases B = {1, x,x?) and B' = {1, x,x2, x3, x4} A=