7. Consider the linear transformation T : (R) → M2x2 (R) defined by ao 2a2 ao- 3a1 4a0 - 12a1 2ao Find the matrix for T, Ts, where 0 00 00 1 are bases for P2R) and M2x2(R) respectively. Find bases for ker(T) and range(T). Is T one-to-one, onto, neither, or both?
2. Let T: P2 P2 be given by T (p(x)) = x2p"(x) – S p(x)dx a. Show that T is a linear transformation b. Find Ker(T) and its basis. Is T one-to-one? c. Find Range(T) and its basis. Is T onto? Verify the dimension theorem.
QUESTION 4 Let T R3-P2 be defined by T(a, b, c) - (a + b + e) +(a+b)a2 (4.1) Show that T is a linear transformation (4.2) Fınd the matrix representation [T]s, B, of T relative to the basıs in R3 and the basis in P2, ordered from left to right Determine the range R(T of T Is T onto? In other words, is it true that R(T)P2 Let x, y E R3 Show that x-y ker(T) f and only...
5. Consider the linear transformation T : P2(R) + Pl(R) defined by T(ax? + bx + c) = (a + b) + (b – c)x. Determine Ker(T), Rng(T), and their dimensions.
Problem 13. Let l be the line in R' spanned by the vector u = 3 and let P:R -R be the projection onto line l. We have seen that projection onto a line is a linear transformation (also see page 218 example 3.59). a). Find the standard matrix representation of P by finding the images of the standard basis vectors e, e, and e, under the transformation P. b). Find the standard matrix representation of P by the second...
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...
Let T. M2(R) →P2(R) be defined by T.(Iga)-(+b) + (b+c) Let T2: P2 (R) → Pl (R) be defined by Tap(x))-p' (x) (c+ d)x2 2. Find Ker(T2 . T) and find a basis for Ker(T2。T).
Tbi b2 Problem 24 : Let b e R4 be a fixed vector, b+0. b3 b4 Define L:R4 → R by 11 12 L(x) = 6-2, x= ER 23 24 where b.x is the dot product of b and 2 in R4. (a) Show that L is a linear transformation. (b) Find the standard matrix representation of L. (c) Find a basis for and the dimension of Ker L. (d) Is L one-to one? Explain why. (e) Is L onto?...
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)
Let a T: M2x2(R) + P2(R), 6 d H (2a +b)x2 + (6 – c)x +(c – 3d). с Let B = 9 (6 8), (8 5), (1 3), ( )) (CO 11),( ( 1),66 1 B' 1 1 ? :-)) C = (x²,2,1) C' = (x + 2,2 +3,22 – 2x – 6). Is T invertible? (1pt) O Yes O No