Hi I have a few questions for you guys. I know that every change of coordinate matrix is invertible is true. Is the converse of this statement also true?
Every invertible matrix is a change of coordinate matrix.
I can't seem to find a counterexample. Also if a polynomial $p(x)$ is an element of $\{P_n\}$ has exactly two terms. Then for any basis $B$ for $P_n$, the coordinate representation $[p]b$ has exactly two non-zero coordinates?
Can you give me a hint on whether or not this is true or false.
If this means a "change of basis matrix", then a matrix is invertible if and only if it is a change of basis matrix. (See e.g. this math.SE question.)
The second part seems to be untrue (if I understand correctly). If we have the polynomial $x+1$ (in the vector space of polynomials of degree $\leq 1$), it has the coordinate representation $(1,1)$ under the basis $\{1,x\}$ and $(0,1)$ under the basis $\{1,x+1\}$.