Written Homework 06

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: Suppose $V$ is a vector space, and assume $\{\bar{v}_1, \bar{v}_2, \bar{v}_3\}$ spans $V$. Also suppose that $\bar{v}_3$ can be written as a linear combination of $\bar{v}_1$ and $\bar{v}_2$. Prove that $\{\bar{v}_1, \bar{v}_2\}$ spans $V$.

• Problem 2: Suppose $V$ is a vector space, and assume $\{\bar{v}_1, \bar{v}_2, \bar{v}_3\}$ is a linearly independent set in $V$. Define $\bar{w} = \bar{v}_1 + \bar{v}_2 + \bar{v}_3$. Prove that $\{\bar{v}_1, \bar{v}_2,\bar{w}\}$ is also linearly independent.