7. Consider the linear transformation T : (R) → M2x2 (R) defined by ao 2a2 ao-...
(9 marks) Consider L: M2x2(R) + P2 (R) defined by a L b d = a + (b + c)2 + dx?. (a) Show that L is a linear transformation, that is, show that L(sA+B) = 8L(A) + tL(B) for any A, B E M2x2(R) and any st ER. (b) Consider p(x) = ao+a1x + a2x2 € P2(R). Show that L is onto by showing that L(A) = P(x) for some matrix A € M2x2(R). Note that you must give...
Let T: P1 → P2 be a linear transformation defined by T(a + bx) = 3a – 2bx + (a + b)x². (a) Find range(T) and give a basis for range(T). (b) Find ker(T) and give a basis for ker(T). (c) By justifying your answer determine whether T is onto. (d) By justifying your answer determine whether T is one-to-one. (e) Find [T(7 + x)]], where B = {-1, -2x, 4x2}.
Suppose T: M2,2 P2 is a linear transformation whose action is defined by s and that we have the ordered bases 1 00 1 0 000 0 00 010 0 1 D-1x2 for M2.2 and P2 respectively. a) Find the matrix of T corresponding to the ordered bases B and D MD(T) 0 0 0 b) Use this matrix to determine whether T is one-to-one or onto < Select an answer >, < Select an answer >
(11) Let the linear transformation T : M2x2(R) + P2 (R) be defined by T (+ 4) = a +d+(6–c)n +(a–b+c+d)a? (1-1) (i) (3 marks) Find a basis for the T-cyclic subspace generated by (ii) (3 marks) Determine rank(T).
0.0KB lll 4G ) 8:06 O Expert Q&A 22. Let T be the linear transformation from Py over R to R22 defined by T (ao+a1x +azx+ax) an-at ai-ar az-a ao + ay Find bases A' of Pa and B' of R2x2 that satisfy the conditions given in Theorem 5.19. Let T be an arbitrary linear transformation of U into V, and let r be the rank of T. Then there exist bases A' of U and B' of V such...
Consider the linear transformation T : R2 + R2 defined as T(21,12)=(0,21 – 12). Find the standard matrix for T: a ab sin(a) 8 f E д 0 0 1 What is the dimension of ker(T)? Is T one-to-one? no 47 Enter one: yes no Write the standard matrix for HoT, where H is the reflection of R2 about the z-axis. a ab sin(a) f 12 II 8 R ат
R2 defined as Consider the linear transformation T: R2 T(21,22)=(0,21 – 22) Find the standard matrix for T: a ab sin (a) f 8 ат What is the dimensi of ker(T)? Is T one-to-one? Enter one: yes no Write the standard matrix for HoT, where H is the reflection of R2 about the 3-axis. a sin(a) f 22 8 R a E är (Alt + A)
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]) =...
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) € R5 | t1 - 3r2 = 0, z3 - 2c4 = 0 and z5 = 0 C5. and its range, R(T), is given by -{1: - -{{:) == ལྟ་ ༢༠༡༧ - R(T) =
ebra MTAS Consider the linear transformation T: R4 R2 defined as T(*1,42,43,44)=(-22 - 3 x3 +2 34,-333 +384). Find the standard matrix for T: sin(a) a Or f 8 R Ω What is the dimension of ker(T)? Is T one-to-one? AY Enter one: yes no Write the standard matrix for HT, where H is the reflection of R2 about the x-axis. ed sin(a) a ax f 8. a Ω