# 威斯康辛大学2210Q 应用线性代数Applied Linear Algebra代考

### 2210Q. Applied Linear Algebra

3.00 credits

Prerequisites: MATH 1132Q or 1152Q or 2142Q. Recommended preparation: a grade of C- or better in MATH 1132Q. May not be taken out of sequence after passing MATH 2144Q, 3210, 3510, or 3710. Repeat restrictions apply. See advising.uconn.edu/repeat-policy for information.

Systems of equations, matrices, determinants, linear transformations on vector spaces, characteristic values and vectors, from a computational point of view. The course is an introduction to the techniques of linear algebra with elementary applications.

## 2210Q 应用线性代数Applied Linear Algebra题目解答

Compute the rank of the linear map $T: \mathbf{R}^{3} \rightarrow \mathbf{R}^{4}$ defined by
$$T(x, y, z)=(x+y+z, x-y, y-z, z-x) .$$

We let $v_{1}, v_{2}, v_{3}$ be the canonical basis of $\mathbf{R}^{3}$ and compute
$$T\left(v_{1}\right)=T(1,0,0)=(1,1,0,-1),$$
thus the first row of the matrix $A$ in the previous discussion is $(1,1,0,-1)$. We do the same with $v_{2}, v_{3}$ and we obtain
$$A=\left[\begin{array}{cccc} 1 & 1 & 0 & -1 \ 1 & -1 & 1 & 0 \ 1 & 0 & -1 & 1 \end{array}\right]$$
Using row-reduction we compute
$$A_{r e f}=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & -1 \ 0 & 0 & 1 & -1 \end{array}\right]$$
and we deduce that
$$\operatorname{rank}(T)=3 .$$

Prove that for all linear transformations $T_{1}, T_{2}: V \rightarrow W$ and all scalars $c \in F$ we have
$${ }^{t}\left(T_{1}+c T_{2}\right)={ }^{t} T_{1}+c^{t} T_{2} .$$

We need to prove that if $l$ is a linear form on $W$, then
$$l \circ\left(T_{1}+c T_{2}\right)=l \circ T_{1}+c l \circ T_{2} .$$
This follows from the fact that $l$ is linear.

$${ }^{t}\left(T_{2} \circ T_{1}\right)={ }^{t} T_{1} \circ{ }^{t} T_{2} .$$
b) Deduce that if $T: V \rightarrow V$ is an isomorphism, then so is ${ }^{t} T: V^{} \rightarrow V^{}$, and $\left({ }^{t} T\right)^{-1}={ }^{t}\left(T^{-1}\right)$.

a) Let $l$ be a linear form on $V_{3}$. Then
$$\begin{gathered} { }^{t}\left(T_{2} \circ T_{1}\right)(l)=l \circ\left(T_{2} \circ T_{1}\right)=\left(l \circ T_{2}\right) \circ T_{1}= \ { }^{t} T_{1}\left(l \circ T_{2}\right)={ }^{t} T_{1}\left({ }^{t} T_{2}(l)\right)={ }^{t} T_{1} \circ{ }^{t} T_{2}(l) . \end{gathered}$$
The result follows.
b) Since $T$ is an isomorphism, there is a linear transformation $T^{-1}$ such that $T \circ$ $T^{-1}=T^{-1} \circ T=$ id. Using part a) and the obvious equality ${ }^{t}$ id $=\mathrm{id}$, we obtain
$${ }^{t} T \circ{ }^{t}\left(T^{-1}\right)=\mathrm{id}={ }^{t}\left(T^{-1}\right) \circ t T,$$
from where the result follows.

Let $A \in M_{2}(\mathbf{C})$ have eigenvalues $\lambda_{1}$ and $\lambda_{2}$. Prove that for all $n \geq 1$ we have
$$\operatorname{Tr}\left(A^{n}\right)=\lambda_{1}^{n}+\lambda_{2}^{n}$$
Deduce that $\lambda_{1}^{n}$ and $\lambda_{2}^{n}$ are the eigenvalues of $A^{n}$.

Let $x_{n}=\operatorname{Tr}\left(A^{n}\right)$. Multiplying relation (2.1) by $A^{n}$ and taking the trace yields
$$x_{n+2}-\left(\lambda_{1}+\lambda_{2}\right) x_{n+1}+\lambda_{1} \lambda_{2} x_{n}=0 .$$
Since $x_{0}=2$ and $x_{1}=\operatorname{Tr}(A)=\lambda_{1}+\lambda_{2}$, an immediate induction shows that $x_{n}=\lambda_{1}^{n}+\lambda_{2}^{n}$ for all $n$.

For the second part, let $z_{1}, z_{2}$ be the eigenvalues of $A^{n}$. By definition, they are the solutions of the equation $t^{2}-\operatorname{Tr}\left(A^{n}\right) t+\operatorname{det}\left(A^{n}\right)=0$. Since $\operatorname{det}\left(A^{n}\right)=(\operatorname{det} A)^{n}=$ $\lambda_{1}^{n} \lambda_{2}^{n}$ and $\operatorname{Tr}\left(A^{n}\right)=\lambda_{1}^{n}+\lambda_{2}^{n}$, the previous equation is equivalent to
$$t^{2}-\left(\lambda_{1}^{n}+\lambda_{2}^{n}\right) t+\lambda_{1}^{n} \lambda_{2}^{n}=0 \quad \text { or } \quad\left(t-\lambda_{1}^{n}\right)\left(t-\lambda_{2}^{n}\right)=0 .$$
The result follows.

## MATH 2210Q MIDTERM EXAM II PRACTICE PROBLEMS

Date and place: Thursday, November 6th, 2021.
Material: Sections 3.1-3.2-3.3-4.1–4.7. Lecture 13–24, HW5-6-7, and the practice exam and additional practice problems below.
Policy: No calculators will be allowed at the exam.
Format: The actual exam will have the same format as the practice problems. The content will
be reasonably similar to the practice exam and the additional practice problems.