Contraction transformation proof

[charmmy]

1. Homework Statement

Q1) Show that a contraction transformation from V to V has a diagonal matrix representation regardless of the basis given to V (same basis in domain and range).

!2) T(x1,x2,x3)=(x1,x2,x2*x3) find the kernel and range T for this transformation.

2. Homework Equations
3. The Attempt at a Solution

Q1)how do we go about building a matrix representation? just arbitarily orr?? Where should I even start?

Q2) the transformation is not linear, so we can't actually find an Echloen matrix to row reduce?

 

[Dick]

You've seen the problem correctly. T isn't linear. So you can't use matrices. You'll have to think more directly about what kernel and range mean. Start with the kernel. How can (x1,x2,x2*x3)=(0,0,0)?

 

[Dick]

Homework Statement

Show that a contraction transformation from V to V has a diagonal matrix representation regardless of the basis given to V (same basis in domain and range).

Homework Equations

The Attempt at a Solution

how do we go about building a matrix representation? just arbitarily orr?? Where should I even start?

You posted a completely different question to begin with. I wish I had quoted it. Then you deleted that question and substituted another. PLEASE DON'T DO THAT. Can you guess how confusing this can be?

 

[fzero]

You build a matrix representation by picking a basis \{v_1,\ldots,v_n\} for V. Then a linear transformation L:V\rightarrow V has a matrix representation L_{mn} by decomposing L(v_m) in terms of the basis according to

L(v_m) = \sum_n L_{mn} v_n.

You should start with the definition of a contraction and apply that to an arbitrary basis vector.

(Dick, if I'm googling correctly, a contraction is a multiplication by a scalar a, with 0<a<1. charmmy should post the definition to verify this.)

Edit: I evidently didn't see the original version of the question...

 

[charmmy]

its only possible if x1 =0; x2=0 and x3=0 right? so its the trivial solution?

 

[Dick]

You build a matrix representation by picking a basis \{v_1,\ldots,v_n\} for V. Then a linear transformation L:V\rightarrow V has a matrix representation L_{mn} by decomposing L(v_m) in terms of the basis according to

L(v_m) = \sum_n L_{mn} v_n.

You should start with the definition of a contraction and apply that to an arbitrary basis vector.

(Dick, if I'm googling correctly, a contraction is a multiplication by a scalar a, with 0<a<1. charmmy should post the definition to verify this.)

Edit: I evidently didn't see the original version of the question...

The original question is back if you look quick. It might change at any moment.

 

[charmmy]

I'ms so sorry about that. I didn't realize you've answered the question before I change it. Can I post both the questions under the same post then?



1. Homework Statement

Q1) Show that a contraction transformation from V to V has a diagonal matrix representation regardless of the basis given to V (same basis in domain and range).

!2) T(x1,x2,x3)=(x1,x2,x2*x3) find the kernel and range T for this transformation.

2. Homework Equations

The Attempt at a Solution

Q1)how do we go about building a matrix representation? just arbitarily orr?? Where should I even start?

Q2) the transformation is not linear, so we can't actually find an Echloen matrix to row reduce?

 

[Dick]

its only possible if x1 =0; x2=0 and x3=0 right? so its the trivial solution?

Don't change the question again, ok? Post a new thread for the other one. No, (x1,x2,x2*x3)=(0,0,0) doesn't just have the trivial solution. There are others.

 

[charmmy]

Okay. sorry about the confusion


fzero: contraction does mean multiplication with a scalar between 0 and 1.

dick: hmm in that case, x3 would be a free variable? since x2 and x1 must be zero.

 

[Dick]

Okay. sorry about the confusion


fzero: contraction does mean multiplication with a scalar between 0 and 1.

dick: hmm in that case, x3 would be a free variable? since x2 and x1 must be zero.

Yes. x3 doesn't matter. So the kernel subspace is (0,0,x3). Now what about the image?

 

[charmmy]

the image would just be everything in R³ is that correct?

 

[Dick]

the image would just be everything in R³ is that correct?

It would seem so to me.

 

[Dick]

It would seem so to me.

Ooops. I take that back. Points like (0,0,1) aren't in the range, are they?