Problem List
The problems to turn in are below. Please organize your work, justify your steps, and write in complete sentences with correct punctuation. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!
- Problem 1: Let $V$ be a vector space with respect to operations $\oplus$ and $\odot$. Using just the axioms in the definition of a vector space, prove that for arbitrary $\mathbf{u},\mathbf{v},\mathbf{w} \in V$, if $\mathbf{u} \oplus \mathbf{v} = \mathbf{u} \oplus \mathbf{w}$, then $\mathbf{v} = \mathbf{w}$.
- Please carefully justify each step by citing which axiom you are using.
- Hint: start like this: “Let $\mathbf{u},\mathbf{v},\mathbf{w} \in V$, and assume $\mathbf{u} \oplus \mathbf{v} = \mathbf{u} \oplus \mathbf{w}$.” Then keep going by using the axioms to manipulate this equation until you arrive at $\mathbf{v} = \mathbf{w}$.
- Problem 2: Let $A$ be an $n\times n$ matrix, and let $\lambda \in \mathbb{R}$ be a scalar. Define $W$ to be the subset of $\mathbb{R}^n$ consisting of all vectors $\mathbf{v} \in \mathbb{R}^n$ satisfying the equation $A\mathbf{v} = \lambda \mathbf{v}$. Prove that $W$ is a subspace of $\mathbb{R}^n$ (with respect to usual vector addition and scalar multiplication on $\mathbb{R}^n$).
- Note: both $A$ and $\lambda$ are arbitrary, so you can not choose what they are. You have to work with them abstractly.
- Hint: you have three things to prove: $W$ is nonempty, $W$ is closed under addition, and $W$ is closed under scalar multiplication. When proving $W$ is closed under addition, start like this: “Assume $\mathbf{w}_1,\mathbf{w}_2 \in W$. Thus $A\mathbf{w}_1 = \lambda \mathbf{w}_1$ and $A\mathbf{w}_2 = \lambda \mathbf{w}_2$.” Now keep going to compute $A(\mathbf{w}_1 + \mathbf{w}_2)$ and show it is equal to $\lambda(\mathbf{w}_1 + \mathbf{w}_2)$.
