1. If a system Ax = b has more than one solution, so does the system Ax = 0.
True. Either a system has no solutions (inconsistent), one solution (consistent with no free variables means unique solution) or an infinite number of solutions (consistent with one or more free variables).
2. If A is mxn and the equation Ax = b is consistent for every b in R^m, then A must have m pivot columns.
True. If A has m pivot columns, then the reduced matrix cannot have a row of zeroes.
3. If none of the R^3 vectors in S = {v1, v2, v3} is a multiple of one of the other vectors, then S is linearly independent.
False. You can still have a linear relation such as v1 = v2 + v3. For example, if v2 = (1,0), v3 = (0,1) and v1 = (1,1), the set S is linearly dependent but none of the vectors is a multiple of the others. (Vectors were written horizontally for typographical convenience.)
4. The equation Ax = 0 has the trivial solution if and only if there are no free variables.
False. Any Ax = 0 has the trivial solution. If you got this wrong, maybe you had it confused with this: Ax = 0 has ONLY the trivial solution if and only if the columns of A are linearly independent.
5. If A is a 5x4 matrix, the linear transformation x -> Ax cannot map R^4 onto R^5.
True. A linear transformation will map onto R^5 if and only if its matrix has 5 pivots. In this case, there cannot be more than 4 pivots because there are only 4 columns.