Question

let let T : R^3 --> R^2 be a linear transformation defined by T ( x, y , z) = ( x-2y -z , 2x + 4y - 2z) a give an example of two elements in K ev( T ) and show that these sum i also an element of K er( T)

Answer #1

**Solution:**

The given linear transformation is defined by T( x, y, z) = ( x - 2y - z , 2x + 4y - 2z). Now let ( x, y, z ) and ( a, b, c ) be two elements in Ker(T). Then we have T(x, y, z ) = 0 and T(a, b, c) = 0 i.e. ( x - 2y - z , 2x + 4y - 2z) = 0 and ( a - 2b - c , 2a + 4b - 2c) = 0 .

Now, T(x+a, y+b, z+c) = ( x+a - 2y - 2b - z - c , 2x +2a + 4y + 4b - 2z - 2c)

= ( x - 2y - z , 2x + 4y - 2z) + ( a - 2b - c , 2a + 4b - 2c)

= 0 + 0 = 0.

This show that (x+a, y+b, z+c) is in Ker(T) i.e. (x, y, z ) + (a, b, c ) is in Ker(T) .

This completess the proof.

** RESULT:** Since Ker(T) is a subspace of
, so sum of any two elements of Ker(T) is in Ker(T). The proof of
this result is given above.

Let T: R^3----> R^3 where T(x,y,z) = (x-2z,y+z,x+2y) . Is T a
one-to-one transformation?
Is the range of T R^3 ? Explain

Consider the mapping R^3 to R^3 T[x,y,z] = [x-2z, x+y-z, 2y]
a) Show that T is a linear Transformation
b) Find the Kernel of T
Note: Step by step please. Much appreciated.

a. Let →u = (x, y, z) ∈ R^3 and define T : R^3 → R^3 as
T( →u ) = T(x, y, z) = (x + y, 2z − y, x − z)
Find the standard matrix for T and decide whether the map T is
invertible.
If yes then find the inverse transformation, if no, then explain
why.
b. Let (x, y, z) ∈ R^3 be given T : R^3 → R^2 by T(x, y, z) = (x...

How many distinct invariant subspaces does the linear operator
T: R^3 --> R^3 defined by T(x,y,z) = (4z-y, x+2z, 3z) have?
0
1
2
3
4

Let
T : P(R) → P(R) be the linear map defined by T(p(x)) = xp′(x) (you
may take it for granted that T is linear). Show that for each λ ∈ Z
with λ ≥ 0, λ is an eigenvalue of T , and xλ is a corresponding
eigenvector.

. In this question we will investigate a linear transformation F
: R 2 → R 2 which is defined by reflection in the line y = 2x. We
will find a standard matrix for this transformation by utilising
compositions of simpler linear transformations. Let Hx be the
linear transformation which reflects in the x axis, let Hy be
reflection in the y axis and let Rθ be (anticlockwise) rotation
through an angle of θ. (a) Explain why F =...

let
T: P3(R) goes to P3(R) be defined by T(f(x))= xf'' (x) + f'(x).
Show that T is a linear transformation and determine whther T is
one to one and onto.

The T: R 4 → R 4 , given by T (x, y, z, w) = (x + y, y, z, 2z +
1) is a linear transformation? Justify that.

Let D be the solid region defined by D = {(x, y, z) ∈ R3; y^2 +
z^2 + x^2 <= 1},
and V be the vector field in R3 defined by: V(x, y, z) = (y^2z +
2z^2y)i + (x^3 − 5^z)j + (z^3 + z) k.
1. Find I = (Triple integral) (3z^2 + 1)dxdydz.
2. Calculate double integral V · ndS, where n is pointing
outward the border surface of V .

Consider the transformation T: R2 -> R3 defined by
T(x,y) = (x-y,x+y,x+2y)
Answer the Following
a)Find the Standard Matrix A for the linear transformation
b)Find T([1
-2])
c) determine if c = [0 is in the range of the transformation
T
2
3]
Please explain as much as possible this is a test question that
I got no points on. Now studying for the final and trying to
understand past test questions.

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