2016-39.md

course	course_year	question_number	tags	title	year
Linear Algebra	IB	39	IB 2016 Linear Algebra	Paper 1, Section II, F	2016

Let $M_{n, n}$ denote the vector space over $F=\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ matrices with entries in $F$. Let $\operatorname{Tr}: M_{n, n} \rightarrow F$ denote the trace functional, i.e., if $A=\left(a_{i j}\right){1 \leqslant i, j \leqslant n} \in M{n, n}$, then

$$\operatorname{Tr}(A)=\sum_{i=1}^{n} a_{i i}$$

(a) Show that Tr is a linear functional.

(b) Show that $\operatorname{Tr}(A B)=\operatorname{Tr}(B A)$ for $A, B \in M_{n, n}$.

(c) Show that $\operatorname{Tr}$ is unique in the following sense: If $f: M_{n, n} \rightarrow F$ is a linear functional such that $f(A B)=f(B A)$ for each $A, B \in M_{n, n}$, then $f$ is a scalar multiple of the trace functional. If, in addition, $f(I)=n$, then $f=$ Tr.

(d) Let $W \subseteq M_{n, n}$ be the subspace spanned by matrices $C$ of the form $C=A B-B A$ for $A, B \in M_{n, n}$. Show that $W$ is the kernel of Tr.