Linear Algebra Hello I need help with regard to these true o

+++++++++++++++++++++++++++Linear Algebra Hello, I need help with regard to these true or false questions. Thank You!!++++++++++++++++++++++++++++

4. The reduced echeon form of a matrix is unique. 5. Whenever a system has free variables, the solution set contains many solutions. 6. Asking whether the linear system corresponding to an augmented matrix [a, an augmented matrix Lij a, \"z b has a solution a2 b] has a solution amounts to asking whether b is in Span|a, az a,) amounts to asking whether b is in Spa,a 7. The weights cr c2, ....cp in a linear combination +G22cpp cannot all be zero.

Solution

4) The statement is TRUE

Since there is only one unique row echleon form form for a particular matrix

5) The statement is FALSE

Reason: The presence of atleast one unique solution for system of variables is independet of presence or absence of free variables, hence the given statement is FALSE

6) The statement is TRUE

Since the set [a1 a2 a3] must be independent in order to have b in their span, since it is part of augmented matrix, hence the b will be spanned by the vectors [a1 a2 a3]

7) The statement is FALSE

They can all be zero in case of the vectors vi all being independent, then the only possible solution will be all the coefficients must be zero