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

1. ## 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 ?

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

So choose any basis of $\displaystyle B_2$ and any 2 matrices in $\displaystyle M_2(\mathbb{R})$ 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

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

So choose any basis of $\displaystyle B_2$ and any 2 matrices in $\displaystyle M_2(\mathbb{R})$ 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 .

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

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

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

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

Tonio

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

So choose any basis of $\displaystyle B_2$ and any 2 matrices in $\displaystyle M_2(\mathbb{R})$ 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 ???

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

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

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

Tonio