Question

If the equation \[Gx = y\] has more than one solution for some y in \[{\mathbb{R}^{\bf{n}}}\] , can the columns of G span \[{\mathbb{R}^{\bf{n}}}\]? Why or why not?

Short answer

The columns of G do not span \[{\mathbb{R}^n}\] because G is not invertible.

Step-by-step solution

Describe the given statement

Given that \[Gx = y\] has more than one solution for some \[y \in {\mathbb{R}^n}\]. This implies that the equation \[Gx = y\] does not have a unique solution for each \[y \in {\mathbb{R}^n}\].

Use the inverse matrix theorem

By the inverse matrix theorem, G is not invertible.

Draw a conclusion

Thus, the columns of G do not span\[{\mathbb{R}^n}\] based on statement (h) of the inverse matrix theorem.