an explicit basis

Author: RobinHankin

vignettes/tensorprod.Rmd

@@ -58,17 +58,17 @@ $$
 T\left(v_1,\ldots, v_i + {v'}_i,\ldots, v_k\right)=
 T\left(v_1,\ldots,v_i,\ldots,v_k\right)+
 T\left(v_1,\ldots,{v'}_i,\ldots,v_k\right),\\
-T\left(v_1,\ldots,av_i,\ldots,v_k\right)=aT\left(v_1,\ldots,v_i,\ldots,v_k\right)
+T\left(v_1,\ldots,av_i,\ldots,v_k\right)=aT\left(v_1,\ldots,v_i,\ldots,v_k\right).
 $$
 
-A multilinear function $T\colon v^k\longrightarrow\mathbb{R}$ is
+A multilinear function $T\colon V^k\longrightarrow\mathbb{R}$ is
 called a *$k$-tensor* on $V$ and the set of all $k$-tensors, denoted
 by $\mathcal{J}^k(V)$, becomes a vector space (over $\mathbb{R}$) if
 for $S,T\in\mathcal{J}^k(V)$ and $a\in\mathbb{R}$ we define
 
 $$
-(S+T)(v_1,\ldots,v_k) = S(v_1,\ldots,v_k) + T(v_1,\ldots,v_k)
-(aS)(v_1,\ldots,v_k) = a\cdot S(v_1,\ldots,v_k)
+(S+T)(v_1,\ldots,v_k) = S(v_1,\ldots,v_k) + T(v_1,\ldots,v_k)\\
+(aS)(v_1,\ldots,v_k) = a\cdot S(v_1,\ldots,v_k).
 $$
 
 There is also an operation connecting the various spaces
@@ -154,4 +154,68 @@ different order.
 rm(T)  # tidyup
 ```
 
+## An explicit basis
+
+
+We will consider an element $X$ of $\mathcal{J}^{2}(V)$ where
+$V=\mathbb{R}^3$ and construct an explicit basis for it along the
+lines of Spivak's observation above.
+
+```{r}
+(X <- ktensor(spray(matrix(c(1,2,3,2,1,1),3,2),1:3)))
+```
+
+Thus $X=\phi_1\otimes\phi_2 +2\phi_2\otimes\phi_1
++3\phi_3\otimes\phi_1$.  Spivak asserts that $\mathcal{J}^{2}(V)$ has
+dimension $n^k=3^2=9$.
+
+In package idiom, for example, $\phi_2\otimes\phi_5$ would be
+constructed as follows:
+
+```{r}
+(phi2_x_phi5 <- ktensor(spray(t(c(1,3)))))
+```
+
+Using this notation we can construct a basis for $\mathcal{J}^{2}(V)$
+as follows:
+
+```{r}
+f <- function(v){ktensor(spray(t(v)))}
+f(c(2,5))
+```
+
+Thus `f()` creates an element of the basis set, in this case
+$\phi_2\otimes\phi_5$.  We can construct $X$ line by line:
+
+```{r}
+1*f(c(1,2)) + 2*f(c(2,1)) + 3*f(c(3,1))
+```
+
+With a little effort, we can create all $3^2=9$ elements of a basis as
+follows:
+
+```{r}
+basis <- apply(expand.grid(rep(list(seq_len(3)),2)),1,f)
+length(basis)
+basis[[6]]
+```
+
+Or it might be better to use ellipsis constructs to pass multiple
+arguments at once:
+
+
+```{r}
+s <- function(...){f(unlist(list(...)))}
+s(3,4,6)
+```
+
+Then we could have 
+
+```{r}
+1*s(1,2) + 2*s(2,1) + 3*s(3,1)
+1*s(1,2) + 2*s(2,1) + 3*s(3,1) == X
+```
+
+
+
 # References