Add start of solutions to 3D

exercises/axler-3d.md

@@ -0,0 +1,53 @@
+# Exercises from Axler 3.C
+
+1, 2, 6, 9, 18, 20
+
+## Question 1
+
+Suppose $T\in\mathcal{L}(V, w)$ is invertible. Show that $T^{-1}$ is
+invertible and $(T^{-1})^{-1} = T$.
+
+### Answer
+
+When asked to show that something has a certain property, it's often
+best to go back to the definition of that property. A linear map
+$R:V\to W$ is invertible (this is definition 3.59) if there exists an
+$S:W\to V$ such that $RS$ is the identity on $W$ and $SR$ is the
+identity on $V$.
+
+So, in order to show that some linear map (say $T^{-1}$) is
+invertible, we must exhibit another linear map with the required
+properties.
+
+The obvious candidate for an inverse of $T^{-1}$ is $T$. Let's check
+whether it is. We need to check that $T^{-1}T$ is the inverse on $V$
+and that $TT^{-1}$ is the inverse on $W$. And indeed they are --
+because _those_ conditions are precisely the meaning of $T^{-1}$
+(which we are assured exists by the question).
+
+## Question 2
+
+Suppose $T\in \mathcal{L}(U, V)$ and $S\in \mathcal{L}(V, W)$ are both
+invertible. Prove that $ST$ is invertible and that $(ST)^{-1} =
+T^{-1}S^{-1}$.
+
+### Answer
+
+To prove that something is invertible we must exhibit its
+inverse. Fortunately, the question has told us what the inverse is, so
+what we need to do is check that it does in fact satisfy the
+conditions for being an inverse.
+
+That is, we need to show: (1) that $\bigl(ST\bigr) (\bigl
+T^{-1}S^{-1}\bigr)$ is the identity on $W$; and (2) that $(\bigl
+T^{-1}S^{-1}\bigr) \bigl(ST\bigr)$ is the identity on $U$.
+
+To show (1), note that composition of linear maps is associative --
+that is, it doesn't matter where we put the parentheses. Thus, roughly
+speaking, we can “cancel the inner $T$s” and then, having got rid of
+those, “cancel the outer $S$s.”
+
+More carefully, $ $\bigl(ST\bigr) (\bigl T^{-1}S^{-1}\bigr) = S(T
+T^{-1})S^{-1}$ because composition is associative. And that is
+$S\mathbf{1}_V S^{-1}$, which is $SS^{-1}$ which is
+$\mathbf{1}_W$. The same argument works for (2).

implementation/README.md

@@ -101,10 +101,17 @@ Using Numpy.
 
 ### Racket
 
-Naive implementation, boxed floats.
+1. `naive`: Boxed floats and generic numeric operations..
+2. `naive-cons`: Replacing `struct` with a pair.
+3. `flo`: Using floating-point * and + in the inner loop of matrix
+   multiply and flvectors. (But still safe.)
+
+All on Apple M2.
+
+| Device       |    s |     ops/s |
+|:-------------|-----:|----------:|
+| `naive`      | 39.4 |   426,272 |
+| `naive-cons` | 26.9 |   623,688 |
+| `flo`        | 11.4 | 1,472,000 |
+| `unsafe`     | 5.23 | 3,208,000 |
 
-| Device       |     s |   ops/s |
-|:-------------|------:|--------:|
-| Intel i7     |       |         |
-| Apple M1 Pro |       |         |
-| Apple M2     | 39.36 | 426,272 |