SOLUTION: I have my final exam at 3 and I was not able to work through this problem on the study guide. Let D: V->V be a linear map defined by D(v)=dv/dt. Find det(D) if V is the space g

Algebra ->  College  -> Linear Algebra -> SOLUTION: I have my final exam at 3 and I was not able to work through this problem on the study guide. Let D: V->V be a linear map defined by D(v)=dv/dt. Find det(D) if V is the space g      Log On


   

Question 250155: I have my final exam at 3 and I was not able to work through this problem on the study guide.
Let D: V->V be a linear map defined by D(v)=dv/dt. Find det(D) if V is the space generated by (e^t,e^2t,e^3t)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Step 1) Apply the linear map D to each vector in the basis to get the vectors e%5Et, 2e%5E%282t%29, and 3e%5E%283t%29.


Step 2) Now let's write the output vectors e%5Et, 2e%5E%282t%29, and 3e%5E%283t%29 as linear combinations of the basis vectors .

So e%5Et=1%2Ae%5Et%2B0%2Ae%5E%282t%29%2B0%2Ae%5E%283t%29, 2e%5E%282t%29=0%2Ae%5Et%2B2%2Ae%5E%282t%29%2B0%2Ae%5E%283t%29 and 3e%5E%283t%29=0%2Ae%5Et%2B0%2Ae%5E%282t%29%2B3%2Ae%5E%283t%29


Step 3)

The coefficients to the linear combinations will form the matrix

%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C2%2C0%2C0%2C0%2C3%29%29


Note: the first column is formed from the coefficients in the first equation, the second from the second equation, etc.


Step 4)

Finding the determinant of a diagonal matrix is trivial since it is simply the product of the diagonal entries. So if

A=%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C2%2C0%2C0%2C0%2C3%29%29


then


det%28A%29=1%2A2%2A3=6


Because the determinant of A is 6, the determinant of the linear transformation D is also 6.