I cant understand some terminology

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In summary, the conversation discusses a transformation T and its application on vectors (1,2) and (3,4). It is mentioned that these two vectors are linearly independent and form a basis of R2, and that (3,4) spans the image of T. It is also stated that the dimension of imT is 1 and the dimension of kerT is 1, making the transformation singular. Lastly, the concept of spanning an image is clarified as being able to represent any element in ImT as a multiple of (3,4) using real numbers.
  • #1
transgalactic
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"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"
??
 
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  • #2
That it spans the image of T means that any element in ImT can be represented as [itex]\gamma*(3,4)[/itex] for some real number [itex]\gamma[/itex]

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:
[tex]T(\vec{v})=\alpha{T}(3,4)+\beta{T}(1,2)=\gamma(3,4), \gamma=\alpha+\beta[/tex]
 
  • #3
thanks
 

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