# Matrix proof - augmented matrix - row reduction - column operation - proof

• Feb 18th 2011, 06:09 AM
maximus101
Matrix proof - augmented matrix - row reduction - column operation - proof
If we let A be the augmented m x (n + 1) matrix of a system of m linear equations
with n unknowns

Let B be the m x n matrix obtained from A by removing the last
column.

Let C be the matrix in row reduced form obtained from A by elementary
row operations.

Prove the the following statements are equivalent:

(i) The linear equations have no solutions

(ii) If \$\displaystyle c_1\$,......, \$\displaystyle c_{n+1}\$ are the columns of A, then \$\displaystyle c_{n+1}\$ is not a linear combination of
\$\displaystyle c_1\$,......, \$\displaystyle c_n\$
• Feb 19th 2011, 04:58 AM
HallsofIvy
Well, what have you done on this? You might think about how the last column of A relates to the first n columns in light of the fact that " A is the augmented m x (n + 1) matrix of a system of m linear equations
with n unknowns".
• Feb 19th 2011, 06:57 AM
maximus101
ok, since it is the augmented matrix then we can get a system of m linear equations, I'm not sure why exactly they are linear but we will use the fact that

A has m rows and each column of A will have c\$\displaystyle _1\$ .... c\$\displaystyle _n\$ we will make our linear equations but I think we can only achieve this if m=n?

these are my first thoughts