| course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
39 |
|
Paper 3, Section II, 9E |
2021 |
(a) (i) State the rank-nullity theorem.
Let $U$ and $W$ be vector spaces. Write down the definition of their direct sum $U \oplus W$ and the inclusions $i: U \rightarrow U \oplus W, j: W \rightarrow U \oplus W$.
Now let $U$ and $W$ be subspaces of a vector space $V$. Define $l: U \cap W \rightarrow U \oplus W$ by $l(x)=i x-j x .$
Describe the quotient space $(U \oplus W) / \operatorname{Im}(l)$ as a subspace of $V$.
(ii) Let $V=\mathbb{R}^{5}$, and let $U$ be the subspace of $V$ spanned by the vectors
$$\left(\begin{array}{c}
1 \\
2 \\
-1 \\
1 \\
1
\end{array}\right),\left(\begin{array}{l}
1 \\
0 \\
0 \\
1 \\
0
\end{array}\right),\left(\begin{array}{c}
-2 \\
2 \\
2 \\
1 \\
-2
\end{array}\right)$$
and $W$ the subspace of $V$ spanned by the vectors
$$\left(\begin{array}{c}
3 \\
2 \\
-3 \\
1 \\
3
\end{array}\right),\left(\begin{array}{l}
1 \\
1 \\
0 \\
0 \\
0
\end{array}\right),\left(\begin{array}{c}
1 \\
-4 \\
-1 \\
-2 \\
1
\end{array}\right)$$
Determine the dimension of $U \cap W$.
(b) Let $A, B$ be complex $n$ by $n$ matrices with $\operatorname{rank}(B)=k$.
Show that $\operatorname{det}(A+t B)$ is a polynomial in $t$ of degree at most $k$.
Show that if $k=n$ the polynomial is of degree precisely $n$.
Give an example where $k \geqslant 1$ but this polynomial is zero.