Linear algebra: midterm

MATH 331 Linear Algebra .... Spring 2013 ......................... NAME_____________

MIDTERM: 6 problems, 10 points each

Instructions:

• Justify all your steps and conclusions.
• For justification you can use all theorems stated in class, unless that's what you are asked to prove. While making references be as specific as possible.
• You cannot use for justification: the results that appeared in homework, quizzes, or other classes.

Problems

1. Suppose $V$ is a vector space with addition that satisfies: $v+w=0$ for all $v,w\in V$. Use only the axioms to prove that $V=\{0\}$.
2. Suppose $W$ is a vector space and suppose $U,V$ are its subsets. Suppose also that $U$ and $V$ are closed under scalar multiplication of $W$. Prove or disprove: (a) $U \cap V$ is closed under scalar multiplication, (b) $U \cup V$ is closed under scalar multiplication.
3. Find the reduced row echelon form of the following system of linear equations. What kind of set is its solution set? $$-x-2y+z=0$$ $$3x\;\;\;\;\;\;\;\;\;\;+z=2$$ $$x-y+z=1$$
4. Suppose $S$ is a subspace of $V$ and $\dim S=\dim V$. Prove that $S=V$.
5. Suppose $V$ is the vector space of all continuous functions. Represent in the standard form the line in $V$ that passes through $\sin$ and $\cos$.
6. Represent the system of linear equation in problem #3 as a matrix equation.