I cant understand some terminology

[transgalactic]

"T(1,2)=(3,4)
T(T(1,2))=T(3,4)=(3,4)

(1,2) and (3,4) are linearly independant and form a basis of R2, so (3,4) spans ImT. Therefore dim(imT)=1, and obviously dim(kerT)=1, so the transformation is singular."

this is a solution to some problem
i can't understand what is "spans an image"
i got
T(1,2)=(3,4)
and
T(3,4)=(3,4)

i know that the image is the resolt of putting the vector in the transformation

but what meens

"(3,4) spans ImT"
??

 

[arildno]

That it spans the image of T means that any element in ImT can be represented as $\gamma*(3,4)$ for some real number $\gamma$

Now, to prove this, an ARBITRARY vector in R2 can be represented as:
[tex]\vec{v}=\alpha(3,4)+\beta(1,2)[/itex] since those two are linearly independent.

Therefore using the properties of a linear transformation, we get:
$$T(\vec{v})=\alpha{T}(3,4)+\beta{T}(1,2)=\gamma(3,4), \gamma=\alpha+\beta$$

 

[transgalactic]

thanks