Let there be a linear transformation going from <mi mathvariant="double-struck">R 3

kokoszzm 2022-06-06 Answered
Let there be a linear transformation going from R 3 to R 2 , defined by T ( x , y , z ) = ( x + y , 2 z x ). Find the transformation matrix if base 1:
( 1 , 0 , 1 ) , ( 0 , 1 , 1 ) , ( 1 , 0 , 0 ) ,
base 2: ( 0 , 1 ) , ( 1 , 1 )
An attempt at a solution included calculating the transformation on each of the bases in R 3 , (base 1) and then these vectors, in their column form, combined, serve as the transformation matrix, given the fact they indeed span all of B 1 in B 2
Another point: if the basis for R 3 and R 2 are the standard basis for these spaces, the attempt at a solution is a correct answer.
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Answers (1)

Hadley Cunningham
Answered 2022-06-07 Author has 20 answers
Let us call the basis of R 3 { λ 1 , λ 2 , λ 3 } and the basis of R 2 { γ 1 , γ 2 }
So all you need to do now is the following:
T ( λ 1 ) = α 11 γ 1 + α 21 γ 1
T ( λ 2 ) = α 12 γ 1 + α 22 γ 1
T ( λ 3 ) = α 13 γ 1 + α 23 γ 1
while α i , j R
And the matrix will look like:
[ T ] = ( α 11 α 12 α 13 α 21 α 22 α 23 )
And this is how it is done
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