Linear Algebra Questions

Hi,

I need step by step solutions for all the 5 questions in the attached file. Needs to be completed within 1 hour.

Thanks 🙂

VI
Ildl
20
160%
2 3
Math 231, Linear Algebra, Exam 2
YOU MUST SHOW ALL DETAILS OF YOUR ARGUMENTS TO RECEIVE CREDIT
1. Let V and W be vector spaces and f:V + W be a linear map. Give the definition of
Ker f and show that f is one-to-one if and only if Ker f = {0} .
2. Using Gaussian elimination, calculate showing all details and explaining all of the
steps:
(a)
– 1
(23)
(b)
9 6 3
det | 8 5 2
7 4 1
3. Does there exist a 3 x 2 matrix whose rank is equal to 3 and at most 1 entry is equal to
zero? If yes, find such a matrix. If not, justify why not. Do not forget to fully justify
your answer.
4. Let P2 = {azx2 + ax + ao | ao, ai, a2 € R} and f : P2 → Rºbe defined by
f(p) = ()
3.
{0 (7)}
where p (1) is the value of the derivative of p at 1 (similarly for p'(2)). Find the matrix
of f if the basis of P2 is {x?, 1, 2} and the basis of Ra is
5. Let V be a vector space (not necessarily R”), f :V + V a linear map, v an eigenvector
of f with eigenvalue 231, and w an eigenvector of f with eigenvalue 2020. Prove
using the definition of eigenvectors and eigenvalues that the vectors v, w are linearly
independent.