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

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) (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 , , and .

Step 2) Now let's write the output vectors , , and as linear combinations of the basis vectors

.

So , and

Step 3)

The coefficients to the linear combinations will form the matrix

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

then

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