Ok for the first question, two lts B and C are equivalent iff there exist lts E and F such that
B = E^-1 C F
Now let E = P and let F=Q, we have
B= P^-1 C Q or PB = CQ so this means that the lts P and Q must be invertible?
Homework Statement
1) two linear transformations B and C are equivalent iff there exist invertible linear transformations P and Q such that PB=CQ
2) if A and B are equivalent then so are A' and B' in dual space
3) Do there exist linear transformations A and B such that A and B are equivalent...
Homework Statement
How could I prove that the subset of a finite dimensional space is also finite dimensional?
Homework Equations
N/A
The Attempt at a Solution
I think it's more intuitive in the sense that since the vector space is finite dimensional the subset is forcibly finite...
Homework Statement
Define linear functionals lk on Pn by
lk(p) = p(tk)
Show that the lk are linearly independent.
Homework Equations
q1(t)= (t-t2)(t-t3)...(t-tn)
The Attempt at a Solution
I really don't know where to start. I do have the following equation:
If the lks are...
Homework Statement
The vectors x1 = (1,1,1), x2=(1,1,-1), x3= (1, -1, -1) form a basis of C3. If {y1, y2, y3} is the dual basis, and if x= (0,1,0), find [x,y1], [x,y2]] and [x,y3].Homework Equations
N/A
The Attempt at a Solution
Ok, first I know that y1= (1,1,1), y2= (1,1,-1) and y3= (1,-1,-1)...
Ok, so let x=(\xi1,\xi2,\xi3) (where \xi1=\xi2) and let y be the functional such that y = \xi1+\xi2+0*\xi3
So for xi (i from 0 to n) y(xi) would equal 0.
Homework Statement
Define a non-zero linear functional y on C^2 such that if x1=(1,1,1) and x2=(1,1,-1), then [x1,y]=[x2,y]=0.
Homework Equations
N/A
The Attempt at a Solution
Le X = {x1,x2,...,xn} be a basis in C3 whose first m elements are in M (and form a basis in M). Let X' be...