Linear Transformations, Change of Basis and Conjugation
Let V = Q3 and let ' be the linear transformation from V to itself:
'(x, y, z) = (9x + 4y + 5z,−4x − 3z,−6x − 4y − 2z), x, y, z E Q
With respect to the standard basis B find the matrix representing this linear transformation.
Take the basis E = {(2,−1,−2), (1, 0,−1), (3,−2,−2)} = {ei} for V and find the matrix representing ' with respect to this basis, i.e. MEE ('). Find the matrix P that conjugates... into ...
Determine whether the following are linear transformations from C[0,1] into R^1.
L(f) = |f(0)|
L(f) = [f(0) + f(1)]/2
L(f) = {integral from 0 to 1 of [f(x)]^2 dx}^(1/2)
Thanks so much. :)
Are the following examples linear transformations from p3 to p4? If yes, compute the matrix of transformation in the standard basis of P3 {1,x,x^2} and P4 {1,x,x^2,x^3}.
(a) L(p(x))=x^3*p''(x)+x^2p'(x)-x*p(x)
(b) L(p(x))=x^2*p''(x)+p(x)p''(x)
(c) L(p(x))=x^3*p(1)+x*p(0)
1) Define a linear transformation....
a) Find a basis for Ker T.
b) Find a basis for Im T.
c) Is T an onto map?
d) Is T a one-to-one map?
2) Define a linear transformation...
a) Find the matrix for T with respect to the standard basis.
b) Find the matrix for T with respect to { ( ) , ( ) , ( ) } as the basis for R and t
Suppose S is a linear space defined below. Are the following mappings L linear transformations from S into itself? If answer is yes, find the matrix representations of formations (in standard basis):
(a) S=P4, L(p(x))=p(0)+x*p(1)+x^2*p(2)+X^3*p(4)
(b) S=P4, L(p(x))=x^3+x*p'(x)+p(0)
(c) S is a subspace of C[0,1] formed by
Please show the complete steps.
1. Let T: P2 -> P1 be the linear transformation defined by T(p(x)) = p'(x) + p(x). Show that T is linear.
2. Find the matrix for the transformation T given in problem 1 with respect to the standard basis {1, x, x^2}. Then, find all eigenvalues and corresponding eigenvectors for T.
3. True
R stands for the field of real numbers. C stands for the field of complex numbers.
1. Let T be a linear transformation from the set P2(R) of all polynomials of degree at most 2 into itself.
T: P2(R) --> P2(R), given by T(f) = f' - f'', fEP2(R),
where f' is the first and f'' is the second derivative of f.
(a) Find the null
1) Let u=(2,3,0), and v=(-1,2,-2). Find
a) ||u + v||
b) ||u|| + || v ||
c) Find two vectors in R³ with norm 1 orthogonal to be both u and v
d) Find norm of vector u / || u ||
2) For which values of t are vectors u = (6, 5, t), and v = (1,t,t) orthogonal?
3)
a) Find the standard matrix [T] for the linear tra
B1) This question concerns the following two subsets of :
(a) Show that , and find a vector in that does not belong to T. [3]
(b) Show that T is a subspace of . [4]
(c) Show that S is a basis for T, and write down the dimension of T. [7]
(d) Find an orthogonal basis for T that contains the vector .
