Ok so I am stick on three proofs for my linear algebra final adn help on any of all of them would really help with my studying
For the first 2 assume that A is an nxn matrix
1.If the collumns of A span Rn then the homogenous system Ax = 0 has only the trivial solution
2. If the collumns of A are linearly independent, then the columns of A span Rn
These 2 have to be proved without referencing other parts of the invertible matrix theorem
3. Be able to prove: If A is an nxn matrix then lambda is an eigencalue of A if and only if det(A-lamda*In) = 0
The Attempt at a Solution
the first two i have a better idea at than the third, since its an nxn i know that if it spans Rn then it has a pivot in every row and coincidently in every row and therefore every column. This same fact can be used to explain number 2 with them being linearly independent. My problem with these two is that Im having trouble since I cant reference them being part of the Invertible matrix theorem.
For the third one I think I am on the right track but not sure
The determinat of A-IL must equal zero because the determinant of A is simply the product of the eigenvalues. If you replace each eigenvalue into the determinat one at a time and multiply it by I, one of the entries will be replaced by zero and any other vlues multiplied by zero will result in zero. As a result, the determinat must equal zero.
Any help would be great