How to get the 4th equation using this data(lenear algebra)

[transgalactic]

this is the data that i was given about the question:
http://img23.imageshack.us/img23/6041/49022610ps1.th.gif

i need to find the matrix of T
i know that T is of 2X4 size
so i built T as
a1 a2 a3 a4
b1 b2 b3 b4

and multiplied by the given vector
i built a matrix for "a" parameters and "b" parameters
but i got only 3 equation instead of the needed 4
in order to get the values of "a"'s and "b"'s

how to use the independent vectors data to construct the 4th equation??

 

[Mark44]

Let's call your vectors u1 = [1 1 0 1], u2 = [0 1 0 1], and u3 = [1 1 0 0].
Since Tu1 = Tu2 = Tu3 = [1 1], then
Tu1 - Tu2 = [0 0] ==> T(u1 - u2) = [0 0]
Tu1 - Tu3 = [0 0] ==> T(u1 - u3) = [0 0]
Tu2 - Tu3 = [0 0] ==> T(u2 - u3) = [0 0]

These equations tell us that u1 - u2, u1 - u3, and u2 - u3 are in the nullspace of T.
Find these vectors, and find a subset of them that is a linearly independent set. From those vectors you can get values for the entries of the matrix corresponding to the transformation T.

 

[transgalactic]

i solved it differently
i said
T=(a1 a2 a3 a4)
(b1 b2 b3 b4)

and built an equations T*U1=(1,1) T*U2=(1,1) T*U3=(1,1)
so i get for "a" parameters only 3 equations
which is not enough
how to use the data that those vectors are independent
in order to build the 4th "a" parameter equation

( the same thing goes for "b" parameters)

??

 

[Mark44]

You say you solved it differently, but you're still asking questions, so it doesn't sound like the problem is actually solved. On the other hand, I actually found a matrix A for which A*u1 = [1 1], A*u2 = [1 1] and A*u3 = [1 1], so which is the better solution?

 

[transgalactic]

you are not using he fact that two vectors are independent
??

 

[Mark44]

Yes, I am. Look at what I said in post #2.