I don't get this question (T-invariant)

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March 26th 2010, 05:14 PM
Revolutionofidentity

I don't get this question (T-invariant)

Create a non-trivial linear operator T with a kernel containing more than the zero vector for which {k (some symmetric matrix) | k belong R} is a t-invariant space in B2 ( symmetric matrices of size 2x2) )

Would someone please rewrite this question for me in more details, or a better clarification ?
March 27th 2010, 01:33 AM
tonio

Quote:

Originally Posted by Revolutionofidentity

Create a non-trivial linear operator T with a kernel containing more than the zero vector for which {k (some symmetric matrix) | k belong R} is a t-invariant space in B2 ( symmetric matrices of size 2x2) )

Would someone please rewrite this question for me in more details, or a better clarification ?

It seems to be like this: denote $B_2:=\{A\in M_2(\mathbb{R})\;\;\;M^t=M\;\}$ , and you have to define a linear operator $T:B_2\rightarrow B_2$ s.t. $\ker T\neq\{0\}$ ...piece of cake!

So choose any basis of $B_2$ and any 2 matrices in $M_2(\mathbb{R})$ and a third one being the zero matrix, and send the vectors of the basis to these three chosen matrices, and extend by linearity...

Tonio
March 27th 2010, 06:14 AM
Revolutionofidentity

Quote:

Originally Posted by tonio

It seems to be like this: denote $B_2:=\{A\in M_2(\mathbb{R})\;\;\;M^t=M\;\}$ , and you have to define a linear operator $T:B_2\rightarrow B_2$ s.t. $\ker T\neq\{0\}$ ...piece of cake!

So choose any basis of $B_2$ and any 2 matrices in $M_2(\mathbb{R})$ and a third one being the zero matrix, and send the vectors of the basis to these three chosen matrices, and extend by linearity...

Tonio

Hi thanks a lot for the reply. it helped but i'm confused about something, in your guidelines i don't see where i'm gonna use my t-invariant space, it was given in specific numbers.
K[[3,7],[7,-4]], that was a t-invariant in B2.
And by the way, how to "Thank" you ? i don't know how to do that .
March 27th 2010, 07:11 AM
Defunkt

Quote:

Originally Posted by Revolutionofidentity

And by the way, how to "Thank" you ? i don't know how to do that .

http://img710.imageshack.us/img710/950/27749476.jpg
March 27th 2010, 08:51 AM
tonio

Quote:

Originally Posted by Revolutionofidentity

Hi thanks a lot for the reply. it helped but i'm confused about something, in your guidelines i don't see where i'm gonna use my t-invariant space, it was given in specific numbers.
K[[3,7],[7,-4]], that was a t-invariant in B2.
And by the way, how to "Thank" you ? i don't know how to do that .

Ok, so then the map is from $M_2(\mathbb{R})$ to itself, and then do as before: choose a basis $\{A_1,A_2,A_3,A_4\}$ and any 3 matrices $S_1,S_2,S_3\in B_2$ , and map as follows:

$A_i\mapsto S_i\,,\,\,i=1,2,3\,,\,\,A_4\mapsto 0=$ the zero matrix , and extend this by linearity.

This maps $M_2(\mathbb{R})$ to $B_2$ ; if you want a more general thing do choose a basis for $B_2$ , extend it to a basis of all $M_2(\mathbb{R})$ and map as above when you map the basis of $B_2$ to 2 matrices of $B_2$ and to the zero matrix .

Tonio
March 27th 2010, 09:30 AM
Revolutionofidentity

Quote:

Originally Posted by tonio

It seems to be like this: denote $B_2:=\{A\in M_2(\mathbb{R})\;\;\;M^t=M\;\}$ , and you have to define a linear operator $T:B_2\rightarrow B_2$ s.t. $\ker T\neq\{0\}$ ...piece of cake!

So choose any basis of $B_2$ and any 2 matrices in $M_2(\mathbb{R})$ and a third one being the zero matrix, and send the vectors of the basis to these three chosen matrices, and extend by linearity...

Tonio

Hi Man, it worked !! Thanks a lot, and i picked my t-invariant space to be the random M2 matrix. picked a basis, and picked a B2 matrix that's not trivial, and made it belong to the kernel of T. And Made a linear combination then used axioms of L transformation and it worked. I checked by throwing a random symmetric 2x2 matrix, then by throwing a matrix from the T-invariant space, and the transformation belonged to it.

Thanks !

but i have a question, why does it work ???
March 27th 2010, 09:31 AM
Revolutionofidentity

thanks to God i could solve that question, i really need to pass this assignment in order to pass the whole class.
March 27th 2010, 09:45 AM
tonio

Quote:

Originally Posted by Revolutionofidentity

thanks to God i could solve that question, i really need to pass this assignment in order to pass the whole class.

"Thanx to god"? Hmmm....anyway, it works because we applied a very basic and important theorem in finite dimensional linear algebra:

Theorem: If $\{v_1,\ldots,v_n\}$ is a basis of a linear space $V$ , and if $\{w_1,\ldots,w_n\}$ are ANY n vectors in some linear space $W$ defined over the same field as $V$ , then

there exists a unique linear transformation $T:V\rightarrow W$ s.t. $Tv_1=w_i\,\,\,\forall\,i=1,\ldots,n$

Tonio