Linear Algebra Concept question Theory Any set of with fewer

Linear Algebra Concept question:

Theory: Any set of with fewer n vectors cannot span R3. Could you please prove this why this is true by constructing a 3X4 matrix, not in row echelon form, whose columns do not span R3. Please provide an explanation to each step. Thanks

Solution

given p vectors in R^n
, then the corresponding matrix A has p columns
and n rows. If p < n, then the REF of A must have rows of zeros (we can have
at most p leading variables, so some rows have no leading variables). But, if
the REF of A has rows of zeros, the vectors do not span R^n