Linear Algebra Coordinate Systems

Question

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.

It seems that this should be true. What is your definition of a change-of-coordinate matrix? — Ben Grossmann, Jan 28 '14 at 02:30
n vectors (α1, …, αn) with the property that every vector in the space can be expressed uniquely as a linear combination of the basis vectors — user123204, Jan 28 '14 at 02:34
@user123204: the columns of any invertible matrix form a basis. Do you see how to answer your question? — zcn, Jan 28 '14 at 02:55

score 1 · Answer 1 · edited Apr 13 '17 at 12:21

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 $\{1,x+1\}$.

Linear Algebra Coordinate Systems

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