Please put the solution in the form of a formal proof, Thank You.
First will find matrix of T,
By(1)(2)(3)
Rank of [T] is 2 .
And basis for Range(T) is
Null(T) is the set of solution space of
which is equivalent to
dim(Null(T))=1
Basis for Nll(T) is
(c)
Rank nullity theorem,
Let T: R3-R2 be the linear map given by a 2c (a) Find a basis of the range space. (Be sure to jus...
Consider the linear map T: M2,2 → R3 defined by [26] = (a-d, b+c, a+b) Find either the nullity or the rank of T and then use the Rank Plus Nullity Theorem to find the other: nullity(T) = rank(T) -
2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector v in the orthogonal complement of the space 0 Span 2,2 Do not simply compute the cross product. (c) Let A be a 5 × 2 rnatrix with linearly independent columns. Using the rank-nullity theorem applied to AT, and any other results from the course, find the dinension of Col(A) 2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector...
linear algebra Let T: R3 R3 be a linear transformation. Use the given information to find the nullity of T. rank(T) = 1 nullity(T) = Give a geometric description of the kernel and range of T. The kernel of T is the single point {(0, 0, 0)}, and the range of T is all of R3. O The kernel of T is all of R3, and the range of T is the single point {(0, 0, 0)}. The kernel of...
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?...
2. Let [8 Marks] 1 2 -1 1 3 -2 a) Find the null space of the matrix A and determine its dimension b) Find the range of the matrix A and determine rank(A) c) State the rank-nullity theorem and verify that it is valid for the matrix A. 2. Let [8 Marks] 1 2 -1 1 3 -2 a) Find the null space of the matrix A and determine its dimension b) Find the range of the matrix 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 T. R3 R3 be a linear transformation. Use the given information to find the nullity of T. rank(7) - 1 nullity(T) - Give a geometric description of the kernel and range of T. The kernel of T is a plane, and the range of T is a line. o The kernel of T is all of R3, and the range of T is all of R. The kernel of T is the single point {(0, 0, 0)), and the...
Let T be a linear map from R3[z] to R2[z] defined as (T p)(z) = p'(z). Find the matrix of T in the basis: 4 points] Let T be a linear map from Rals] to R12] defined as (TP)(z) = p,(z). Find the matrix of T in the basis: in R2[-]; ~ _ s, r2(z) (z-s)2 in R2 [2], where t and 8 are real numbers. T1(2 Find coordinates of Tp in the basis lo, 1, 12 (if p is...
Let T : R3 → R2 be a linear map. Recall that the image of T, Im(T), is the set {T(i) : R*) (a) Suppose that T(v)- Av. Describe the image of T in terms of A Using this description, explain why Im(T) is a subspace of R2. (b) What are the possible dimensions of Im(T)? (c) Pick one of the possible dimensions and construct a specific map T so that Im(T) has that dimension.
I need the answer to problem 6 Clear and step by step please Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations 1. Show that. TU is also a linear transformation. 2. Show that aT is a linear transformation for any scalar a. 3. Suppose that T is invertible. Show that T-1 is also a linear transformation. Problem 5. Let T : R3 →...