Practice Midterm q.7
Can someone explain what this question is asking? what basis is (a,b) in? Question: Let β={(1,1),(1,-1)} and γ={(1,2),(1,0)} be bases for R2. Suppose that a certain linear transformation T has matrix representation [T] from β to γ = | 1 1 | | 3 2 | Calculate T(a,b) for arbitrary (a,b)
Noeland: May 3, 2015, 1:20 a.m.
My opinion (could be wrong):It asks us to find a linear transformation within one basis. For example, T((a,b))=(b,a) could be written as the same matrix |0 1| | 1 0 |as long as we only consider one basis. However, it is a different matrix if we consider two bases. In this question the matrix(linear transformation) between two bases is given and we should find the transformation within one basis.
davidabraham: May 4, 2015, 12:35 p.m.
So (a,b) is in the β basis, the transformation does some sort of move and converts it to the γ basis, then we convert it back to the β basis?
dyana: May 4, 2015, 3:38 p.m.
Is the answer T(a,b)= {7a-b, 4a}
tedallen: May 4, 2015, 11:29 p.m.
I got {3.5a + .5b, 2a}
jiahengsong: May 5, 2015, 9:58 a.m.
I got {3.5a + .5b, 2a}. When a vector in Rn is given in the "bare" ordered pair or tuple form (such as "(1,1)") without specifying a basis, I think [[1,0],[0,1]] is usually implied to be the basis. With that in mind, we are given [T]β->γ, which takes a vector in R2 expressed in the basis β and gives a result in (a different) R2, expressed in the basis γ. The "input" and "output" R2 should be considered different spaces; however, because they are both R2, both of them can use bases that have the same form, i.e. the basis α=={[1,0],[0,1]} can mean either {[1,0],[0,1]} in the "input" R2, or {[1,0],[0,1]} in the "target" R2. (a,b) is not in the β basis, but in the basis α={[1,0],[0,1]}. Then the question can be restated as follows: given the transformation [T]β->γ, and the identity transformations [I]α->β and [I]α->γ, find [T]α->α.
brian_li: May 5, 2015, 4:44 p.m.
@jiahengsong what would be the equation to find [T]α->α? Would it be ([I]α->β)([T]β->γ)([I]y->α)?
jiahengsong: May 5, 2015, 9:29 p.m.
Yes, although according to the convention for the meaning of rows and columns, the multiplication should be written from right to left: ([I]y->α)([T]β->γ)([I]α->β)
jiahengsong: May 5, 2015, 9:32 p.m.
Just testing... \textsuperscript{a}\textsubscript{b} \textsuperscript{a}\SB{b}
jiahengsong: May 5, 2015, 9:33 p.m.
Sorry... didn't properly escape the expression... $\textsuperscript{a}\textsubscript{b}$
jiahengsong: May 5, 2015, 9:34 p.m.
That sucks. $x^{1}_{y}$
jiahengsong: May 5, 2015, 9:35 p.m.
So by "[T]a->b", what I mean is this: $[T]^{b}_{a}$