Thread: Linearly independent set

If A is a 3x3 matrix and {v1, v2, v3} is a linearly independent set of vectors in R^3 for which {Av1, Av2, Av3} is also a linearly independent set, then the matrix A must be invertible.

Is this true? Can someone please explain why or why not??

I have read a lot but could not find any resources related to this theorem.
I do not have any idea of how to look at this problem.

Originally Posted by mola

If A is a 3x3 matrix and {v1, v2, v3} is a linearly independent set of vectors in R^3 for which {Av1, Av2, Av3} is also a linearly independent set, then the matrix A must be invertible.

Is this true? Can someone please explain why or why not??

I have read a lot but could not find any resources related to this theorem.
I do not have any idea of how to look at this problem.

If the image of a basis is again a basis the transformation (or matrix) must be an isomophism (invertible), since its kernel is zero. Check it.
Of course, the above applies for finite dimensions...

Tonio