Changing the basis of a vector or matrix

Question

I have a quick question. I am aware how to determine a vector with respect to a non-standard basis B, if I have a vector in standard basis and this non-standard basis B.

My question is: If I have a vector in a non-standard basis M, and wish to convert it to the non-standard basis B, do I need to first convert it to the standard basis E, and only then to non-standard basis B?

Answer 1

Yes exactly, we need to use the standard basis, that is

• $v_E=Mv_M$
• $v_E=Bv_B$

then

• $Mv_M=Bv_B \implies v_M=M^{-1}Bv_B$