Final answer:
The matrix A that represents the usual inner product in R2 relative to the given basis vectors can be found using the dot product.
Step-by-step explanation:
To find the matrix A that represents the usual inner product in R2 relative to the basis u1=(1,2) and u2=(-1,3), we need to use the formula:
A = [u1·u1, u1·u2; u2·u1, u2·u2]
where · represents the dot product.
Using the given basis vectors:
u1=(1,2) and u2=(-1,3)
we can calculate:
u1·u1 = (1,2)·(1,2) = 1*1 + 2*2 = 1+4 = 5
u1·u2 = (1,2)·(-1,3) = 1*(-1) + 2*3 = -1+6 = 5
u2·u1 = (-1,3)·(1,2) = -1*1 + 3*2 = -1+6 = 5
u2·u2 = (-1,3)·(-1,3) = -1*(-1) + 3*3 = 1+9 = 10
Therefore, the matrix A that represents the usual inner product is:
A = [5, 5; 5, 10]