06-240/Term Test

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The Test

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Math 240 Algebra I - Term Test

University of Toronto, October 24, 2006

Solve the 5 problems on the other side of this page.

Each of the problems is worth 20 points.

You have an hour and 45 minutes.

Notes.

• No outside material other than stationary and a basic calculator is allowed.
• We will have an extra hour of class time in our regular class room on Thursday, replacing the first tutorial hour.
• The final exam date was posted by the faculty - it will take place on Wednesday December 13 from 2PM until 5PM at room 3 of the Clara Benson Building, 320 Huron Street (south west of Harbord cross Huron, home of the Faculty of Physical Education and Health).

Good Luck!

Questions Page

Solve the following 5 problems. Each of the problems is worth 20 points. You have an hour and 45 minutes.

Problem 1. Let $F$ be a field with zero element $0_{F}$, let $V$ be a vector space with zero element $0_{V}$ and let $v\in V$ be some vector. Using only the axioms of fields and vector spaces, prove that $0_{F}\cdot v=0_{V}$.

Problem 2.

1. In the field {\mathbb C} of complex numbers, compute ${\frac {1}{2+3i}}+{\frac {1}{2-3i}}$      and     ${\frac {1}{2+3i}}-{\frac {1}{2-3i}}$.
2. Working in the field ${\mathbb {Z} }/7$ of integers modulo 7, make a table showing the values of $a^{-1}$ for every $a\neq 0$.

Problem 3. Let $V$ be a vector space and let $W_{1}$ and $W_{2}$ be subspaces of $V$. Prove that $W_{1}\cup W_{2}$ is a subspace of $V$ iff $W_{1}\subset W_{2}$ or $W_{2}\subset W_{1}$.

Problem 4. In the vector space $M_{2\times 2}({\mathbb {Q} })$, decide if the matrix ${\begin{pmatrix}1&2\\-3&4\end{pmatrix}}$ is a linear combination of the elements of $S=\left\{{\begin{pmatrix}1&0\\-1&0\end{pmatrix}},\ {\begin{pmatrix}0&1\\0&1\end{pmatrix}},\ {\begin{pmatrix}1&1\\0&0\end{pmatrix}}\right\}$.

Problem 5. Let $V$ be a finite dimensional vector space and let $W_{1}$ and $W_{2}$ be subspaces of $V$ for which $W_{1}\cap W_{2}=\{0\}$. Denote the linear span of $W_{1}\cup W_{2}$ by $W_{1}+W_{2}$. Prove that $\dim(W_{1}+W_{2})=\dim W_{1}+\dim W_{2}$.

Good Luck!

Solution Set

Students are most welcome to post a solution set here.

1. $0_{F}\cdot v=(0_{F}+0_{F})\cdot v$ (by F3)

$(0_{F}+0_{F})\cdot v=0_{F}\cdot v+0_{F}\cdot v$ (by VS8)

By VS4, $\exists \ (0_{F}\cdot v)'s.t.(0_{F}\cdot v)+(0_{F}\cdot v)'=0_{V}$

Add $(0_{F}\cdot v)'$ to both sides of $0_{F}\cdot v=0_{F}\cdot v+0_{F}\cdot v$

$(0_{F}\cdot v)'+(0_{F}\cdot v)=[(0_{F}\cdot v)'+0_{F}\cdot v]+0_{F}\cdot v$

$0_{V}=0_{V}+0_{F}\cdot v$ (by construction)

$0_{V}=0_{F}\cdot v$ (by VS3)

2. 1) ${\frac {1}{2+3i}}+{\frac {1}{2-3i}}={\frac {(2-3i)+(2+3i)}{(2+3i)(2-3i)}}={\frac {4}{2^{2}+3^{3}}}={\frac {4}{13}}$

${\frac {1}{2+3i}}-{\frac {1}{2-3i}}={\frac {(2-3i)-(2+3i)}{(2+3i)(2-3i)}}={\frac {-6i}{2^{2}+3^{3}}}={\frac {-6}{13}}i$