Finding the Bases for kernel and range of linear transformation.

[avister51291]

Homework Statement

How do I find the bases for both the kernel and range of this linear transformation?
Let T: R4 ----> R4 be the linear transformation that takes [1101] and [1011] to [2304] and takes [1110] and [0111] to [3120]

a. Find the bases for both the kernel and the range of this linear transformation.
b. Give and orthonormal basis for the kernel of this transformation
c. Indicate the matrix that would represent T under the standard basis for R4

Homework Equations

Gram-Schmidt
u1 = v1
u2 = v2 - (<v,u>/<u,u>)u

The Attempt at a Solution

For part a:
I can find the basis for the kernel but I do not know how to find it for the range.
Basis for Kernel:
Since the transformation takes two vectors to one, I used this property to find out the
basis for the kernel, which is T(u - v) = [1101 -1011] = [0,1,-1,0]
I did the same for the other two vectors which resulted in [1,0,0,-1].

Basis for the Range: ?

For part b:
I took the two basis vectors and used the Gran-Schmidt to find the orthonormal basis for the kernel which is {\sqrt{}2/2 (1,0,0,-1), \sqrt{}2/2(0,1,-1,0)}

for part c: I do not know how to do it. I know how to find a transformation matrix given two transformations that go to different vectors; however, that is not the case here as two vectors transform to the same vector. This happens for two different sets.

 

[Mark44]

Homework Statement

How do I find the bases for both the kernel and range of this linear transformation?
Let T: R4 ----> R4 be the linear transformation that takes [1101] and [1011] to [2304] and takes [1110] and [0111] to [3120]

a. Find the bases for both the kernel and the range of this linear transformation.
b. Give and orthonormal basis for the kernel of this transformation
c. Indicate the matrix that would represent T under the standard basis for R4

Homework Equations

Gram-Schmidt
u1 = v1
u2 = v2 - (<v,u>/<u,u>)u

The Attempt at a Solution

For part a:
I can find the basis for the kernel but I do not know how to find it for the range.
Basis for Kernel:
Since the transformation takes two vectors to one, I used this property to find out the
basis for the kernel, which is T(u - v) = [1101 -1011] = [0,1,-1,0]

You're being very sloppy here. For a linear transformation T, it must be true that T(u - v) = T(u) - T(v).
So T(u - v) = T([1101] -[1011]) = T[0,1,-1,0]
and T(u) - T(v) = [2 3 0 4] - [2 3 0 4] = [0 0 0 0].
So T[0,1,-1,0] = [0 0 0 0].
Is this what you meant to say?

I did the same for the other two vectors which resulted in [1,0,0,-1].

Do you mean to say that T[1,0,0,-1] = [0 0 0 0]?

From this it appears that the ker(T) is at least dimension 2 and maybe more.

Basis for the Range: ?

For part b:
I took the two basis vectors and used the Gran-Schmidt to find the orthonormal basis for the kernel which is {\sqrt{}2/2 (1,0,0,-1), \sqrt{}2/2(0,1,-1,0)}

for part c: I do not know how to do it. I know how to find a transformation matrix given two transformations that go to different vectors; however, that is not the case here as two vectors transform to the same vector. This happens for two different sets.

 

[avister51291]

Yes, for part A I meant that in order to find the basis for the kernel I would need to do
T(u-v) = T(U) - T(V). The vectors that would go to [0 0 0 0] under the transformation are
[0,1,-1,0] and [1,0,0,-1].
So the basis for the kernel of this transformation is B= {[0,1,-1,0], [1,0,0,-1]}
How would I go about finding the basis for the range and transformation matrix by looking at what I was given?

 

[Mark44]

Let's give these vectors some names:
u₁ = [1101]
u₂ = [1011]
u₃ = [1110]
u₄ = [0111]

From your previous work, you know that T(u₁ - u₂) = 0 and that T(u₃ - u₄) = 0.

IOW, you know that T([01-10]) = 0 and T([100-1]) = 0.

What about T(u₁ + u₂)? T(u₃ + u₄)?