MAT223 theorems

UofT MAT223 linear algebra theorems

(1) Row equivalence theorem

If two augmented matrices are row equivalent, the systems they represent have the same solution

(1) Gauss Jordan Theorem

Every matrix A is equialent to some matrix B in row echelon form, furthermore, If a matrix C is in rref, then that matrix is unique

(1) Pivot parameterization lemma

If a system of linear equations has at least one solution, then the pivot variables can be parameterized in terms of the non pivot variables.

(1) Rouche capelli

1) An augmented matrix in rref with a pivot in the final column has no real solutions

2)An Augmented matrix in rref who’s last pivot is in the second last row has exactly one solution

3) An augmented matrix in rref who’s second last column (that is, the last variable column) has no pivot, has infinitely many solutions

(2) Linear dependence theorem

The set of vectors {v1, v2, v3, ….., vn} is linearly dependent if and only if it’s matrix has a non trivial solution (i.e. not all variables are equal to zero) [TESTABLE PROOF]