right now Im studying for my linear algebra exam and I think I pretty much understood the basic concepts but then I tried the following and I am not able to solve it, can someone help me with this:
In the vector space is U a subspace with the following polynomials as basis :
Additionally there are the following functions:
which form a family
a) Calculate the Transformation Matrix and the inverse.
b) Show why is a basis
c) Calculate the Coordinates of
d) For the linear map calculate the Transformation matrices and .
I have no clue how to begin. I think i know how to solve b) and inverting a matrix is not a problem but the rest is not clear to me!
Can you help me, please!
this is not a inner product, this is the symbol we use for the transformation matrix ( ).
What i tried is this:
and my transformationmatrix is
Calculating the inverse is not that a problem. Showing that this is a basis I show that the columns are linear independent.
For c) my idea is to multiply the Matrix with the Vector q:
And for d) i have no idea!
A general method for writing a linear transformation from vector space U to vector space V, given specific bases for U and V, is:]
Apply the linear transformation to each of the basis vector for U in turn. The result will be in V and so can be written as a linear combination of the basis vectors for V. The coeffients in that linear combination will form one column of the matrix.
Here, U and V are the same and you are given two different bases for it so essentially we write the basis "vectors" of the second matrix as linear combinations of the first and use those coefficients as columns of the matrix.
The first basis "vector" is and it is to be transformed into so the first column of the matrix representation is . The second basis "vector" is and it is to be transformed into so the second column of the matrix representation is . I don't see where you got that "-3". There is no "3" in any of the basis vectors.
To find the "inverse" of that, you can do either of two things:
1) Find the inverse matrix to the one you just found.
2) Do the process in "reverse"- write the basis vectors in the first basis as linear combinations of the vectors in the second basis and use the coefficients as columns of the matrix.
Do it both ways as a check.