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How can we prove that all change-of-basis matrices are invertible? The trivial case when it's a change of basis for $\mathbb{R^{n}}$ is easily demonstratable using, for example, determinants. But I am struggling to rigorously show this for all bases, for example for a two-dimensional subspace of $\mathbb{R^{4}}$. I am sure that there are many ways to go about this proof, and I would be very appreciative for as many ways of demonstration as possible, to back up my intuition!

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For something like this you probably should include the definitions that you use. This is because there are different pedagogical presentations of linear algebra that treats different things as definitions and different things as consequences of those definitions. In one of the presentations I am familiar with, a change-of-basis matrix is pretty much by definition invertible (being a square matrix of full rank). –  Willie Wong Mar 8 '11 at 16:57
    
@Willie: in my class we acknowledge the matrix to be square, but I haven't heard the mention of full rank. I guess it would be a consequence of it being a change of basis matrix, but that's not something we spoke of. –  user7961 Mar 8 '11 at 17:01
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I agree with Willie. Off the top of my head, here is one way to look at it (which may or may not agree with your setup): view the change of basis matrix as a linear transformation $L$ which carries one basis $v_1,...,v_n$ to a different basis $w_1,...,w_n$. Then there is a unique linear transformation carrying the basis $w_1,...,w_n$ to $v_1,...,v_n$ and the matrix of this transformation must be the inverse to the first change of basis matrix. –  Pete L. Clark Mar 8 '11 at 17:08
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What is a change-of-basis matrix except a matrix which is invertible? –  Qiaochu Yuan Mar 8 '11 at 17:44
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@Karamislo: A change-of-basis matrix is never a $4\times2$ matrix. Irrespective of whether the vector space is $\mathbb{R}^2$ or a two-dimensional subspace of $\mathbb{R}^4$, as long as it's two-dimensional, all its bases have 2 elements, and any matrix representing a change from one basis to another is a $2\times2$ matrix, to which you can apply exactly the same reasoning as if it represented a change of basis in $\mathbb{R}^2$. –  joriki Mar 8 '11 at 19:08

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up vote 8 down vote accepted

What is a change-of-basis matrix? You have a vector space $\mathbf{V}$ (and it doesn't matter if $\mathbf{V}$ is all of $\mathbb{R}^n$, or some subspace thereof, or even something entirely different), and two different ordered bases for $\mathbf{V}$, $\beta_1$ and $\beta_2$ (necessarily of the same size, since two bases of the same vector space always have the same size): \begin{align*} \beta_1 &= \Bigl[ \mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n\Bigr]\\ \beta_2 &= \Bigl[ \mathbf{w}_1,\mathbf{w}_2,\ldots,\mathbf{w}_n\Bigr]. \end{align*} A "change of basis" matrix is a matrix that translates from $\beta_1$ coordinates to $\beta_2$ coordinates. That is, $A$ is a change-of-basis matrix (from $\beta_1$ to $\beta_2$) if, given the coordinate vector $[\mathbf{x}]_{\beta_1}$ of a vector $\mathbf{x}$ relative to $\beta_1$, then $A[\mathbf{x}]_{\beta_1}=[\mathbf{x}]_{\beta_2}$ gives the coordinate vector of $\mathbf{x}$ relative to $\beta_2$, for all $\mathbf{x}$ in $\mathbf{V}$.

How do we get a change-of-basis matrix? We write each vector of $\beta_1$ in terms of $\beta_2$, and these are the columns of $A$: \begin{align*} \mathbf{v}_1 &= a_{11}\mathbf{w}_1 + a_{21}\mathbf{w}_2+\cdots+a_{n1}\mathbf{w}_n\\ \mathbf{v}_2 &= a_{12}\mathbf{w}_1 + a_{22}\mathbf{w}_2 +\cdots + a_{n2}\mathbf{w}_n\\ &\vdots\\ \mathbf{v}_n &= a_{1n}\mathbf{w}_1 + a_{2n}\mathbf{w}_2 + \cdots + a_{nn}\mathbf{w}_n. \end{align*} We know we can do this because $\beta_2$ is a basis, so we can express any vector (in particular, the vectors in $\beta_1$) as linear combinations of the vectors in $\beta_2$.

Then the change-of-basis matrix translating from $\beta_1$ to $\beta_2$ is $$ A = \left(\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right).$$

Why is $A$ always invertible? Because just like there is a change-of-basis from $\beta_1$ to $\beta_2$, there is also a change-of-basis from $\beta_2$ to $\beta_1$. Since $\beta_1$ is a basis, we can express every vector in $\beta_2$ using the vectors in $\beta_1$: \begin{align*} \mathbf{w}_1 &= b_{11}\mathbf{v}_1 + b_{21}\mathbf{v}_2 + \cdots + b_{n1}\mathbf{v}_n\\ \mathbf{w}_2 &= b_{12}\mathbf{v}_2 + b_{22}\mathbf{v}_2 + \cdots + b_{n2}\mathbf{v}_n\\ &\vdots\\ \mathbf{w}_n &= b_{1n}\mathbf{v}_n + b_{2n}\mathbf{v}_n + \cdots + b_{nn}\mathbf{v}_n. \end{align*} So the matrix $B$, with $$B = \left(\begin{array}{cccc} b_{11} & b_{12} & \cdots & b_{1n}\\ b_{21} & b_{22} & \cdots & b_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ b_{n1} & b_{nn} & \cdots & b_{nn} \end{array}\right),$$ has the property that given any vector $\mathbf{x}$, if $[\mathbf{x}]_{\beta_2}$ is the coordinate vector of $\mathbf{x}$ relative to $\beta_2$, then $B[\mathbf{x}]_{\beta_2}=[\mathbf{x}]_{\beta_1}$ is the coordinate vector of $\mathbf{x}$ relative to $\beta_1$.

But now, consider what the matrix $BA$ does to the standard basis of $\mathbb{R}^n$ (or $\mathbf{F}^n$, in the general case): what is $BA\mathbf{e}_i$, where $\mathbf{e}_i$ is the vector that has a $1$ in the $i$th coordinate and zeros elsewhere? It's a matter of interpreting this correctly: $\mathbf{e}_i$ is the coordinate vector relative to $\beta_1$ of $\mathbf{v}_i$, because $[\mathbf{v}_i]={\beta_1}=\mathbf{e}_i$. Therefore, since $A[\mathbf{x}]_{\beta_1} = [\mathbf{x}]_{\beta_2}$ and $B[\mathbf{x}]_{\beta_2}=[\mathbf{x}]_{\beta_1}$ for every $\mathbf{x}$, we have: $$BA\mathbf{e}_i = B(A\mathbf{e}_i) = B(A[\mathbf{v}_i]_{\beta_1}) = B[\mathbf{v}_i]_{\beta_2} = [\mathbf{v}_i]_{\beta_1} = \mathbf{e}_i.$$ That is, $BA$ maps $\mathbf{e}_i$ to $\mathbf{e}_i$ for $i=1,\ldots,n$. The only way for this to happen is if $BA=I_n$ is the identity. The same argument, now interpreting $\mathbf{e}_i$ as $[\mathbf{w}_i]_{\beta_2}$, shows that $AB$ is also the identity.

So $A$ and $B$ are both invertible.

So every change-of-basis matrix is necessarily invertible.

It doesn't really matter if you are considering a subspace of $\mathbb{R}^N$, a vector space of polynomials or functions, or any other vector space. So long as it is finite dimensional (so that you can define the "change-of-basis" matrix), change-of-basis matrices are always invertible.

Added. I just saw the comment where you give the definition you have of change-of-basis matrix: a matrix $C$ which, when multiplied by a matrix $B$ whose columns form a basis of a certain subspace, produces another matrix $A$ whose columns form a basis for the same subspace.

This matrix $C$ is just the matrix that expresses the columns of $A$ in terms of the columns of $B$. That is, it's the change-of-basis matrix from "columns-of-A" coordinates to "columns-of-B" coordinates.

For example, take the subspace of $\mathbb{R}^4$ given by $x=z$ and $y=w$, with basis $$\left(\begin{array}{c}1\\0\\1\\0\end{array}\right),\quad\left(\begin{array}{c}0\\1\\0\\1\end{array}\right),$$ and now consider the same space, but with basis $$\left(\begin{array}{c}1\\1\\1\\1\end{array}\right),\quad \left(\begin{array}{r}1\\-2\\1\\-2\end{array}\right).$$ The matrix $C$ such that $$ \left(\begin{array}{rr} 1 & 1\\ 1 & -2\\ 1 & 1\\ 1 & -2 \end{array}\right) = C\left(\begin{array}{cc} 1 & 0\\ 0 & 1\\ 1 & 0\\ 0 & 1 \end{array}\right)$$ is obtained by writing each vector in the columns of $A$ in terms of the columns of $B$: \begin{align*} \left(\begin{array}{r} 1\\1\\1\\1\end{array}\right) &= 1\left(\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right) + 1\left(\begin{array}{c}0 \\ 1 \\ 0 \\ 1\end{array}\right),\\ \left(\begin{array}{r} 1\\ -2\\ 1\\ -2\end{array}\right) &= 1\left(\begin{array}{c}1\\0\\1\\0\end{array}\right) -2\left(\begin{array}{c}0\\1\\0\\1\end{array}\right). \end{align*} And so, the matrix $C$ is $$C = \left(\begin{array}{rr} 1 & 1\\ 1 & -2 \end{array}\right).$$ Expressing the columns of $B$ in terms of the columns of $A$ give the inverse: \begin{align*} \left(\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right) &= \frac{2}{3}\left(\begin{array}{c}1 \\ 1\\ 1\\ 1\end{array}\right) + \frac{1}{3}\left(\begin{array}{r}1 \\ -2\\ 1\\ -2\end{array}\right)\\ \left(\begin{array}{c}0\\ 1\\ 0\\ 1\end{array}\right) &= \frac{1}{3}\left(\begin{array}{c} 1\\ 1\\ 1\\ 1\end{array}\right) -\frac{1}{3}\left(\begin{array}{r}1\\ -2\\ 1\\ -2\end{array}\right), \end{align*} so the inverse of $C$ is: $$C^{-1} = \left(\begin{array}{rr} \frac{2}{3} & \frac{1}{3}\\ \frac{1}{3} & -\frac{1}{3} \end{array}\right),$$ which you can verify by multiplying by $C$.

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@Magidin I believe the top right entry of C inverse should be positive. Fantastic explanation! Thanks! –  dandiellie Mar 8 '11 at 20:21
    
@dandielle: thanks! –  Arturo Magidin Mar 8 '11 at 20:24
    
@Arturo Magidin: Nice explanation -- but I don't understand why you brought the standard basis of $\mathbb{R}^n$ into it -- why not just say that since $A$ translates $\beta_1$ coordinates into $\beta_2$ coordinates and $B$ translates $\beta_2$ coordinates into $\beta_1$ coordinates, then applying first $A$ and then $B$ translates $\beta_1$ coordinates into $\beta_2$ coordinates and back to $\beta_1$ coordinates, and thus $BA$ must be the identity matrix (and likewise for $AB$)? –  joriki Mar 8 '11 at 21:28
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@joriki: Actually, now I remember why: Since you are dealing with the matrices $A$ and $B$, you would have to think about the the transformation that maps from $\mathbf{V}$ to $\mathbb{R}^n$ via "coordinate vector", then map to $\mathbb{R}^n$ via $BA$, then map back to $\mathbf{V}$ via "what the coordinate vector means", so I thought it would be a bit more cumbersome than just directly seeing what the matrices $AB$ and $BA$ do to a basis for $\mathbb{R}^n$. –  Arturo Magidin Mar 8 '11 at 23:40
    
@Arturo Magidin: I don't understand. The thought of 'a transformation that maps from $\mathbf{V}$ to $\mathbb{R}^n$ via "coordinate vector"' never crosses my mind when I think about this -- all you need is the fact that the coordinates of a vector in a basis are unique -- then if $BA$ translates from one set of coordinates to another and back, it has to be the identity. To my mind, mentioning "the standard basis of $\mathbb{R}^n$" just confuses things, since it sounds as if $\mathbb{R}^n$ is entering as a vector space, whereas actually it's only entering as the set of n-tuples of coordinates. –  joriki Mar 9 '11 at 5:18

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