For the given linear transformation T : R" - R" determine if (i) T is one-to-one,...
Problem 2. In each part below, either diagonalize the given linear transformation, if possible, or else explain why this is impossible. (That is, find a basis B such that the coordinate matrix [T\B or explain why no such basis exists.) (а) Т: Р2 —> Р2 given by T(p) — ар'. (b) Т:P, — P2 given by T(р) — р(2л — 1). (c) T R2x2 R2x2 given by T(A) = A+ AT. (d) T: С +С given by T(a + bi)...
Determine if there exists a linear transformation T: R2 -> R2 with the following properties. If yes, give an example. If no, explain why such a transformation is not possible. (4) Determine if there exists a linear transformation T: R2 + R2 with the following properties. If yes, give an example. If no, explain why such a transformation is not possible. (a) T is one-to-one and onto. (b) T is not one-to-one. (c) T is not onto. (d) T is...
Determine whether the linear transformation T is one-to-one and whether it maps as specified. Let T be the linear transformation whose standard matrix is 37 1 -2 A=-1 3 -4 -2 -9 Determine whether the linear transformation T is one-to-one and whether it maps R onto R O One-to-one; onto R O Not one-to-one: onto O Not one-to-one; not onto OOne-to-one: not onto
Problem 2. In each part below, either diagonalize the given linear transformation, if possible, or else explain why this is impossible. (That is, find a basis B such that the coordinate matrix [T\B or explain why no such basis exists.) (а) Т: Р2 —> Р2 given by T(p) — ар'. (b) Т:P, — P2 given by T(р) — р(2л — 1). (c) T R2x2 R2x2 given by T(A) = A+ AT. (d) T: С +С given by T(a + bi)...
Determine whether the linear transformation is one-to-one, onto, or neither T: R^2 -> R^2 , T(x,y) = (x-y,y-x)
13. A linear transformation T takes Nº into f". T[ +y - y 2.1 + 3y y (a) Is T one to one? Justify your answer. If not, then give two vectors with the same image. (b) Is T onto? Justify your answer. If not then give a vector in R? that is not an image.
Determine whether or not the following transformation T :V + W is a linear transformation. If T is not a linear transformation, provide a counter example. If it is, then: (i) find the nullspace N(T) and nullity of T, (ii) find the range R(T) and rank of T, (iii) determine if T is one-to-one, (iv) determine if T is onto. : (a) T: R3 + R2 defined by T(x, y, z) = (2x, y, z) (b) T: R2 + R2...
Recall that if T: R" R" is a linear transforrmation T(x) = [Tx, where [T is the transformation matrix, then 1. ker(T) null([T] (ker(T) is the kernel of T) 2. T is one-to-one exactly when ker(T) = {0 3. range of T subspace spanned by the columns of [T] col([T) 4. T is onto exactly when T(x) = [Tx = b is consistent for all b in R". 5. Also, T is onto exactly when range of T col([T]) =...
Suppose T:R4_R4 is the transformation given below. Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by providing two vectors that have the same image under T. If T is not onto, show this by providing a vector in R4 that is not in the range of T. 2x0+6x1+6x2+4x3 -2x0–x1-x2 + x3 |-3x0-8x1-5x2+4x3 xo+5x1+6x2+7x3 2 Tis one-to-one Tis onto
Question 7 Determine whether the linear transformation T is one-to-one and whether it maps as specified. 2 + 3x 3) T(X 1, X 2, x 3) = (-2x 2 - 2x 3, -2x 1 + 8x 2 + 4x 3, -X 1 - 2x 3,3x Determine whether the linear transformation T is one-to-one and whether it maps R 3 onto R4. Not one-to-one; not onto R4 One-to-one; onto R4 Not one-to-one; onto R4 One-to-one; not onto R4