In Axler's "Linear Algebra Done Right", he gave the singular-value decomposition as:

$Tv = s\_1\langle v,e\_1\rangle f\_1 + \cdots + s\_n\langle v,e\_n\rangle f\_n$, where 
T is an operator; $s\_1,\ldots,s\_n$ are the singular values of $T$; $(e\_1,\ldots,e\_n)$ and $(f\_1,\ldots,f\_n)$ are orthonormal bases of $V$. Singular values of $T$ are defined as eigenvalues of $\sqrt{T^\*T}$. $\langle ,\rangle $ is inner product.

I am having trouble connecting this to the more generally used matrix representation $A=UΣV^T$. 
Matrices in Axler's book are considered as linear maps, "with respect to some bases". How do I understand Axler's exposition using matrices?

Thank you. 

**Update:** 
To clarify, I was thinking from the perspective where an operator is a matrix with respect to some basis (in his book...). So an operator can be written as $M(T,(b1,...bn),(b1,...bn))$. If I apply change of basis, I get $M(I,(f1,...fn),(b1,...bn))\*M(T,(e1,...en),(f1,...fn))\*M(I,b1,...bn),(e1,...en))$ 
where (e1,...en) is an orthonormal basis; (f1,...fn) is another orthonormal basis given by $fi=Sei$ given by isometry $S$ from polar decomposition.

$M(T,(e1,...en),(f1,...fn))$ gives me the diagonol matrix $Σ$. But how about the rest? Thanks.

**Update 2**:
Thanks for the answers below and I understand those. But my question is a bit specific to Axler's definition. In Axler's book, he thinks of matrix a bit differently: m\*n matrix is defined as $M(T,(v\_1...v\_n),(w\_1..w\_m))$ where $T$ is a linear map from space $V$ to W, and $(v\_1...v\_n)$ and $(w\_1...w\_m)$ are bases for 2 spaces. For the $k$th column in the matrix, each entry $a\_{ik}$ is defined as $Tv\_k=a\_{1k}w\_1+...a\_{mk}w\_m$, so the matrix represents the linear map between the bases. So it's like "you don't talk about a matrix without talking about $T$ and the bases". Whether this is a good way to think about matrix is probably also a topic for debate...

Anyway, now with that, I am trying to decompose: $M(T,(b\_1,...b\_n),(b\_1,...b\_n))$. I already know that the diagonol $Σ$ in SVD can be thought of as: **A.** $M(T,(e\_1,...e\_n),(f\_1,...f\_n))$, (where $f\_i=Se\_i$ given by isometry $S$ from polar decomposition $T=S\sqrt{T^\*T}$), **OR**, **B.** $M(\sqrt{T^\*T},(e\_1,...e\_n),(e\_1,...e\_n))$. So I would also like to know: 1. the operator 2. the bases, for the other 2 unitary matrices, no matter which one you choose for $Σ$.

Thanks again.