proving if such transformation exists

is there a transformation which follows these rules:

T:$\displaystyle R^4$->$\displaystyle R^4$

T(-1,1,0,1)=(1,-1,2,1)

T(1,1,1,0)=(2,3,1,-1)

T(-1,5,2,3)=(3,7,0,-3)

ImT is defined as span of the base of the images

so i found that this group is dependant

{(1,-1,2,1),(2,3,1,-1),(3,7,0,-3)}

so ImT=Sp{(1,-1,2,1),(2,3,1,-1)}

so dim ImT=2 and dim kerT=2

so i took $\displaystyle R^4$ basis

and defined T as:

T(-1,1,0,1)=(1,-1,2,1)

T(0,2,1,1)=(0,5,-3,-3)

T(0,0,1,0)=(0,0,0,0)

T(0,0,0,1)=(0,0,0,0)

is it ok?

Re: proving if such transformation exists

Quote:

Originally Posted by

**transgalactic** is there a transformation which follows these rules:

T:$\displaystyle R^4$->$\displaystyle R^4$

T(-1,1,0,1)=(1,-1,2,1)

T(1,1,1,0)=(2,3,1,-1)

T(-1,5,2,3)=(3,7,0,-3)

You mean "linear transformation"- you make a general transformation that does anything!

Quote:

ImT is defined as span of the base of the images

so i found that this group is dependant

You mean "set". In "mathematical" English, "group" has a very specific meaning that is not correct here.

Quote:

{(1,-1,2,1),(2,3,1,-1),(3,7,0,-3)}

Note that {(-1, 1, 0, 1), (1, 1, 1, 0), (-1, 5, 2, 3)} are **also** dependent. We can write

-3(-1, 1, 0, 1)+ 2(1, 1, 1, 0)= (-1, 5, 2, 3).

Is T(-1, 5, 2, 3)= -3T(-1, 1, 0, 1)+ 2T(1, 1, 1, 0)?

Quote:

so ImT=Sp{(1,-1,2,1),(2,3,1,-1)}

so dim ImT=2 and dim kerT=2

so i took $\displaystyle R^4$ basis

I don't understand what you mean by this. **What** did you take as the basis? Are you referring to the standard basis?

Quote:

[tex]and defined T as:

T(-1,1,0,1)=(1,-1,2,1)

T(0,2,1,1)=(0,5,-3,-3)

T(0,0,1,0)=(0,0,0,0)

T(0,0,0,1)=(0,0,0,0)

is it ok?

Well, you haven't answered the question! What is your answer?

Re: proving if such transformation exists

a transformation of a basis is single.

so if we write the transformation of a each vector in the R^4 basis

then it defines a single transformation

T(x,y,z,t)=xT(1,0,0,0)+yT(0,1,0,0)+zT(0,0,1,0)+tT( 0,0,0,1)

ok there is no problem for me to find the value of

T(0,0,0,1) etc..

i ask if it ok to constract a transformation in such a way?

how to know if it follows tle laws given in the question

how do you see a definition of transformation?