Do two column equivalent systems share the same solution set?

Question

We know that if two system of linear equations are row equivalent then they share the same solution set,but can we say the same for column equivalence as well?i.e.given a system $Ax=b$and another system $Cx=d$ such that $[A|b]$and $[C|d]$are column equivalent,can we conclude that they have the same solution set?

Answer 1

Note that $[A|\lambda b]$ is column equivalent to $[A|b]$ for any nonzero scalar $\lambda.$ This should be apparent that it breaks solution sets.

For a simple [if somewhat degenerate] example, assuming $A = b = 1$ (i.e. $A, x, b \in \mathbb{R}$) gives $x = 1,$ but this is column equivalent to $x = 2$ by choosing $\lambda = 2.$

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If you want, you can look at every "elementary column operation" to answer this question more generally. It should be apparent that none of them maintains all solution sets.