Question relating to homogeneous system, subspaces and bases.

[mottov2]

Homework Statement

Let A be an m x n matrix such that the homogeneous system Ax=0 has only the trivial solution.

a. Does it follow that every system Ax=b is consistent?

b. Does it follow that every consistent system Ax=b has a unique solution?

The Attempt at a Solution

So if the homogeneous system has only the trivial solution, then according to column space criterion, the zero vector is in the column space of matrix A.
If the zero vector is in the column space of A, then any vector b can be expressed uniquely as a linear combination of column vectors of A.
Hence every consistent system Ax=b has a unique solution.

Im trying to understand all the theories and connect them so I'm not even sure if this makes sense at all.

 

[lanedance]

Homework Statement

Let A be an m x n matrix such that the homogeneous system Ax=0 has only the trivial solution.

a. Does it follow that every system Ax=b is consistent?

b. Does it follow that every consistent system Ax=b has a unique solution?

The Attempt at a Solution

So if the homogeneous system has only the trivial solution, then according to column space criterion, the zero vector is in the column space of matrix A.

the zero vector is trivial and in any subspace...
I would think more about linear independence