Question

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

Answer #1

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)

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.

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.

(1) Consider the linear operator T : R2 ! R2 defined by
T
x
y
=
117x + 80y
??168x ?? 115y
:
Compute the eigenvalues of this operator, and an eigenvector for
each eigen-

Solve
a. x + y + z = 2, x – y + z = 3,
x + y + 2z = 0
b. 5x + y – 2z = 2, x + 2y + 3z
= 2, 2x – y = 3

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

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...

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 .

1)T F: All (x, y, z) ∈ R 3 with x = y + z is a subspace of R 3
9
2) T F: All (x, y, z) ∈ R 3 with x + z = 2018 is a subspace of R
3
3) T F: All 2 × 2 symmetric matrices is a subspace of M22. (Here
M22 is the vector space of all 2 × 2 matrices.)
4) T F: All polynomials of degree exactly 3 is...

Solve each system by elimination.
1) -x-5y-5z=2
4x-5y+4z=19
x+5y-z=-20
2) -4x-5y-z=18
-2x-5y-2z=12
-2x+5y+2z=4
3) -x-5y+z=17
-5x-5y+5z=5
2x+5y-3z=-10
4) 4x+4y+z=24
2x-4y+z=0
5x-4y-5z=12
5) 4r-4s+4t=-4
4r+s-2t=5
-3r-3s-4t=-16
6) x-6y+4z=-12
x+y-4z=12
2x+2y+5z=-15

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