Problem 3. Let V and W be vector spaces, let T : V -> W be a linear transformation, and suppose U is a subspace of W (a) Recall that the inverse image of U under T is the set T-1 U] := {VE V : T(v) E U). Prove that T-[U] is a subspace of V (b) Show that U nim(T) is a subspace of W, and then without using the Rank-Nullity Theorem, prove that dim(T-1[U]) = dim(Unin (T))...
:| Let T : P → R , such that T (ao +ax+a2x2 +a3r)-4 +ai +a, +a3 . a) Prove that T is a linear transformation b) Find the rank and nullity of T. c) Find a basis for the kernel of T.
:| Let T : P → R , such that T (ao +ax+a2x2 +a3r)-4 +ai +a, +a3 . a) Prove that T is a linear transformation b) Find the rank and nullity of T. c) Find a...
Let T: R3 → R3 be the linear transformation that projects u onto v = (9, -1, 1). (a) Find the rank and nullity of T. rank nullity (b) Find a basis for the kernel of T.
Let ai, a2 , аз, bị, b2P3 R. Define T : R3 R2 by Prove T is a linear transformation.
Question 5 For each given vector b and matrix A, determine if b e im(A) 1 -2 3 (a) b 0 A 21 3 0 5 15 (b) b A2-24 9 Question 6 Find the rank and nullity of the given linear transformations T and determine which are one-to-one and which are onto. r+ y ri+r2 Question 7 Find nullity(T) if (a) T:R R2, rank(T) 1 (b) T:RR, rank(T) 0 (c) T : Rs ? R2, rank(T)-1 Question 8 Let...
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...
2. (8 marks] Consider the linear transformation T:R3 R2 TX,Y, 2) = (+y-2, -1-y+z). (a) Show that the matrix (TE.Es representing T in the standard bases of R3 and R² is of the form TEE 1 -1 1 -1 -1 1 (b) Find a basis of the null space of T and determine the dimension of this space. (c) Find a basis of the range of T and determine the dimension of the range of T. (d) Is T Onto?...
7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for the nullspace (Kernel) of T. c) Find a basis for the range of T.
7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for...
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...
Please put the solution in the
form of a formal proof, Thank You.
Let T: R3-R2 be the linear map given by a 2c (a) Find a basis of the range space. (Be sure to justify that it spans and is linearly independent.) (b) Find a basis of the null space. (Be sure to justify that it spans and is linearly independent.) (c) Use parts (a) and (b) to verify the rank-nullity theorem.
Let T: R3-R2 be the linear map...