B = {b1,b2,b3} is a basis for a vector space V and T: V→ V is a linear transformation such that [T]B =
1 |
-1 |
0 |
2 |
-2 |
-1 |
0 |
-1 |
-3 |
Since [T]B =
[T(b1),T(b2), T(b3)] , hence T(b1) = (1,2,0)T, T(b2) = (-1,-2,-1)T and T(b3) = (0,-1,-3)T.
Further, T being a linear transformation , preserves both the vector addition and scalar multiplication. Hence T( -3b1-b2-b3) = -3T(b1)-T(b2)-T(b3) = -3(1,2,0)T -(-1,-2,-1)T -(0,-1,-3)T = ( -2, -3, 4)T.
Find the matrix of the linear transformation T: V →W relative to B and C. Suppose B = {bı, b2, b3} is a basis for V and C = {C1, C2} is a basis for W. Let T be defined by T(b]) = 261 + C2 T(62) = -501 +502 T(b3) = 2C1-802 2. 3 0 2 -6 [3 0 -6 1 5-8 2 -5 2 5 -8 2 1 -5 5 2 -8
7. Let V be the space generated by the basis B = {sin(t), cos(t), et}. i.e. V = span(B). Consider the linear transformation T:V + V defined by T(f(t)) = f"(t) – 2f'(t) – f(t). Find the standard matrix of the transformation. (Hint: Associate sin(t) with the vector (0), and so forth.) 8. Show that B = {t2 – 2, 3t2 +t, t+t+8} is a basis for P2, and find the change of coordinates matrix P which goes from B...
Assume that the transition matrix from basis B = {b1, b2, b3} to basis C = {c1, c2, c3} is PC,B = 1/2*[ 0 -1 1 ; -1 1 1 ; 1 0 0 ]. (a) If u = b1 + b2 + 2b3, find [u]C. (b) Calculate PB,C. (c) Suppose that c1 = (1, 2, 3), c2 = (1, 2, 0), c3 = (1, 0, 0) and let S be the standard basis for R 3 . (i) Find...
10. (10 pt) Let bi1 The set B tbi,b2) is a basis for R2. Let T : R2 →R2 is a linear transformation such that T(bi) 7bi +7b2 and T (b2) 3bi +4b2 Then the matrix of T relative to the basis B is and b2- -1 4 [T]B and the matrix of T,relative to the standard basis E for R2 is
0 1 Let S span 1 1 1 0 }, a basis for S. Show that| (a) Let B1 { 1 0 1 1 0 is also a basis for S 0 B2 { 1 (b) Write each vector in B2 (c) Use the previous part to write each vector in B2 with respect to Bi (how many components should each vB, vector have?) (d) Use the previous part to find a change of basis matrix B2 to B1. What...
How was the linear transformation of b1 and b2 were applied (L(b1) , L(b2))? NOTE: b1=(1,1)^T , b2=(-1,1)^T Linear Transformations EXAMPLE 4 Let L be a linear transformation mapping R? into itself and defined by where (bi, b2] is the ordered basis defined in Example 3. Find the matrix A represent- ing L with respect to [bi, b2l Solution Thus, A0 2 onofosmation D defined by D(n n' maps P into P, Given the ordered Linear Transformations EXAMPLE 4 Let...
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?...
(b) Let E = {(1, C2, C3} be the standard basis for R3, B = {bų, b2, b3} be a basis for a vector space U, and S: R3 → U be a linear transformation with the property that S(X1, X2, X3) (x2 + x3)b1 + (x1 + 3x2 + 3x3)b2 + (-3X1 - 5x2 - 4x3)b3. Find the matrix F for S relative to E and B. INSTRUCTIONS: 1. Use the green arrows next to the answer spaces below...
Problem 24 : Let b b2 b3 ba E R4 be a fixed vector, b + 0. Define L:R4 R by C1 12 L(x) = b. I, 2= ER4. 13 24 where bºx is the dot product of b and x 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...
5. Consider a vector space V = {00 + 0 + ar? + 13. 1 40,, 02, 03 € R} and a linear map D:V+V. P(1) - P(1) (a) Write down the matrix representation L of the linear map D under the basis B, = (b) Consider the new basis B2 = 31.5 +5.368.479 10 Find the matrix transformation T such that B, BT. (c) Let L, be the matrix representation of D under the new basis B. Prove L2...