What does $T^2$ mean if T is a linear transformation?

Question

Sorry if this is a dumb question, but I came across the notation $T^2$ in a work and not sure how to interpret what this means. I've never come across this in my own notes, nor can I find it with alot of googling. In this case, T is a linear transformation

My first idea was just that it means $T \circ T$ but I'm not sure that even makes sense.

Thank you

Answer 1

If $P=\sum_{k=0}^na_kX^k\in\mathbb{K}[X]$ is a polynomial and $T:E\to E$ is a linear transformation on a $\mathbb{K}$-vector space, you can define $P(T)$ as a linear transformation from $E$ to $E$ by $$P(T)=\sum_{k=0}^na_kT^k,\text{ with }T^k=\underbrace{T\circ\dots\circ T}_{k \text{ times}} \text{ for $k\geq1$ and }T^0=\mathrm{Id}_E. $$

Answer 2

If $T$ is a linear transformation from a vector space $V$ to itself (written $T \colon V \to V$), then $T^2$ just means $T \circ T$. Similarly, $T^3 = T \circ T \circ T$, etc.

However, if $T$ is a linear transformation between different vector spaces (written $T \colon V \to W$ with $V \neq W$), then $T \circ T$ does not make sense. In that case, I don't know what could be meant by $T^2$.

Answer 3

Yes, $T^2$ does indeed refer to $T \circ T$. In fact, in the context of linear transformations, composition is often written like this; if $T : V \to W$ and $S : W \to X$ are linear, we often denote $S \circ T$ by $ST$.

I don't have a source for this, but I'm fairly certain this comes from the conventions of matrices. Matrices can be identified with linear transformations in a very natural way, and matrix multiplication is denoted like this.

Answer 4

You've got the right idea. So for example if we have that $T^2=T$ you can show that these linear transformations are projections giving us a useful geometric interpretation of an algebraic expression. Nilpotent transformations will have the property that for so me natural number $k$ that $T^k=0$ and they will be of interest algebraically as well. You'll be able to make certain polynomial expression make sense this way and derive many properties of the transformation if we know it satisfies a certain polynomial equation. This will show up when studying the characteristic and minimal polynomials of a matrix. The only trick it to ensure the composition makes sense and endomorphisms provide a large class of examples in this theme.