I am trying to get a better grasp on invertibility of matrices. My current mental model is basically that, a Matrix is invertible if every value of b can be mapped back to a unique x. Do I have the right idea here?
Is the inverse of the linear transformation $T(x) = b$ equivalent to saying $T^{-1}(b) = x$?
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linear-algebra
matrices
inverse
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1Yes, "invertible" means exactly the same thing as an invertible function. Using your own words (emphasis mine): *Every* value of $b$ (this requirement is basically surjectivity) can be mapped back to a *unique* $x$ (this is injectivity). – 2011-09-23
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2Yes you do, more specifically a linear transformation $T$ can be invertible if it maps $T : V\rightarrow W$, where $V$ and $W$ have the same dimension and is an isomorphism. To be invertible it simply means that we have some $S$ such that $S\circ T = T\circ S$ is the identity transformation. Of course the matrix $[S\circ T] = [T]\times[S]$ – 2011-09-23
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0awesome thanks for the comments everyone! – 2011-09-23
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0But be careful,suppose you have $T = \pmatrix{1 &0\\0 &1\\0&0}$ and $b=Tx$ then you can again obtain $x$ from $b$. And $T$ is not invertible. Hence, you have to pick your wording slightly more rigorously. – 2011-09-23
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0Notice that, $T^T T = I$ almost an inverse but not exactly. The other requirement is that for every $b$ you can find an $x$ such that $b=Tx$. – 2011-09-23
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0@percusse isn't this simply noting that the map needs to be an isomorphism (is bijective)? The ability to "obtain $x$ from $b$" is saying that it is one-to-one and "for every $b$ you can find an $x$ such that $b = Tx$" means it is onto. – 2011-09-23
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0@DevenWare You are of course right. I have missed the *unique* word in the original statement. – 2011-09-23
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1@Deven Ware: I might be wrong, but are you sure that the matrix representation of $[S \circ T]$ is $T \times S$, not the other way around? – 2011-09-23
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0@Gerben Heh you're right; I was thinking first apply $T$ then apply $S$ and ended up writing them in the wrong order, good catch. If someone has the ability to edit old comments, they can change it – 2011-09-23