- #1
LocationX
- 147
- 0
Let V be the vector space of all real integrable functions on [0,1] with inner product [tex]<f,g>=\int_0^1 f(t)g(t)dt[/tex]
Three linear operators defined on this space are [tex] A=d/dt [/tex] and [tex]B=t[/tex] and [tex]C=1[/tex] so that [tex]Af=df/dt[/tex] and [tex]Bf=tf[/tex] and [tex]Cf=f[/tex]
I need to find the eigenvalues of these operators:
For A:
[tex]\frac{df}{dt} = \lambdaf[/tex]
[tex]\frac{d ln(f)}{dt} = \lambda [/tex]
[tex]d(ln(f))=\lambda dt+c[/tex]
[tex]f=ce^{\lambda t}[/tex]
So the eigenvalues for A are continuous and can be any real number.
For B:
[tex]tf=\lambda f[/tex]
[tex]\lambda =t [/tex]
The eigenvalues are the variable t?
For B;
[tex]Cf=f=\lambda f[/tex]
The eigenvalue is 1.
Confused about B.
Three linear operators defined on this space are [tex] A=d/dt [/tex] and [tex]B=t[/tex] and [tex]C=1[/tex] so that [tex]Af=df/dt[/tex] and [tex]Bf=tf[/tex] and [tex]Cf=f[/tex]
I need to find the eigenvalues of these operators:
For A:
[tex]\frac{df}{dt} = \lambdaf[/tex]
[tex]\frac{d ln(f)}{dt} = \lambda [/tex]
[tex]d(ln(f))=\lambda dt+c[/tex]
[tex]f=ce^{\lambda t}[/tex]
So the eigenvalues for A are continuous and can be any real number.
For B:
[tex]tf=\lambda f[/tex]
[tex]\lambda =t [/tex]
The eigenvalues are the variable t?
For B;
[tex]Cf=f=\lambda f[/tex]
The eigenvalue is 1.
Confused about B.