Please answer this question by typing or writing clearly with explanations
Please answer this question by typing or writing clearly with explanations 3. Define T: M >R...
3. [20 marks] A linear transformation T: P2 + R’ is defined by [ 2a – b 1 T(a + bt + ct?) = a +b – 3c LC-a ] (1). [6 marks] Determine the kernel Ker T of the transformation T and express it in the form of a span of basis. Further, state the dimension of Ker T (2). [6 marks) Find the range Range T of the transformation T and express the range in the form of...
3. In the following question, we are going to prove that ker(T) = { } if and only if T is one-to- one. (Writing prove is like writing a little essay, with some good logical connection between each sentence.) (a) Let T:V - W a linear transformation between two vector spaces. Suppose ker(T)={0}. Show that T is one-to-one. (Hint: proof by contradiction, by assuming both ker(T)=ð and T is not one-to-one. Now, apply definition of kernel and one-to-one, what is...
Consider the the transformation T: P^2 -> T^2 defined by T(a+bx+cx^2) = (a+b, b-c). Find the kernel of T. Give 2 examples of vectors in the kernel.
Find a matrix M such that the linear transformation T:R5 → R4 defined by T(x) = Mx has the property that its kernel, ker(T), is given by ker(T) {1: ER5 @1 - 3c2 = 0, c3 - 2c4 = 0 and c5 and its range, R(T), is given by R(T) - {(:) - ༠ ༠ ༠ ༡ e R4 | u + c + + ཀྱ =
Q4 (b) Prove that ker P = ker Tn ker S. () h a Question 4. Define T: Ma2 = c+ d. Prove that T is a linear transformation R by T C and onto. Find dim(ker T). Is T one-to-one? Jamomials in P. Show (b) Prove that ker P = ker Tn ker S. () h a Question 4. Define T: Ma2 = c+ d. Prove that T is a linear transformation R by T C and onto. Find...
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]) =...
Define the linear transformation T by T(X) = Ax. 1 -1 3 A = 0 1 3 1. (a) Find the kernel of T. (If there are an infinite number of solutions use t as your parameter.) ker(T) = (b) Find the range of T. {(-t, t): t is any real number} O R² O {(t, 3t): t is any real number} R {(3t, t): t is any real number}
QUESTION 1. §1.9 THE MATRIX OF A LINEAR TRANSFORMATION Le t T R be the linear transformation defined by t-th AnSwer Find the standard matrix of T. Is T one to one? Is T onto? Jushif'cahon
let T: P2 --> R be the linear transformation defined by T(p(x))=p(2) a) What is the rank of T? b)what is the nullity of T? c)find a basis for Ker(T)
a. 6. Let T: R* → P2(R)be defined as T 2) = (a - 2d) + (c + 3b)x + (a - 2c)x Ld] I Find a basis for the Ker(T). (3pts) b. Find a basis for the Range(T) (3pts) c. Determine whether T is one-to-one. (2pts) d. Determine whether T is onto. (2pts)