Show that the family $\left({u_1}, \ldots ,{u_m},v \right)$ is linearly dependent iff $\left( {u_1}, \ldots ,{u_m} \right)$ is linearly dependent.

Question

I'm having problems with the following problem:

Let $V$ be a vector space over the field $F$. Let $U$ be a subspace of $V$ and let $v \in V \setminus U$. Let $m \in \mathbb{N}$ and let ${u_1}, \ldots ,{u_m}$ be elements of $U$.

1. Show that ${\lambda _1}{u_1} + \ldots + {\lambda _m}{u_m} + \mu v = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0}$ implies that $\mu=0$.

2. Deduce that the family $\left({u_1}, \ldots ,{u_m},v \right)$ is linearly dependent if and only if the family $\left( {u_1}, \ldots ,{u_m} \right)$ is linearly dependent.

For 1.: Should I start out by assuming that $\left( {u_1}, \ldots ,{u_m} \right)$ is linearly independent, and then that $\left( {u_1}, \ldots ,{u_m},v \right)$ is no longer linearly independent, therefore linearly dependent. This means there are scalars $\mu, \lambda_1, \ldots, \lambda_m$ not all equal to zero such that ${\lambda _1}{u_1} + \ldots + {\lambda _m}{u_m} + \mu v = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0}$ . Where do I go from there?

For 2.: I believe that I should first prove

If $\left({u_1}, \ldots ,{u_m},v \right)$ is linearly dependent, then the family $\left( {u_1}, \ldots ,{u_m} \right)$ is linearly dependent.

Then also prove the converse:

If the family $\left( {u_1}, \ldots ,{u_m} \right)$ is linearly dependent, then $\left({u_1}, \ldots ,{u_m},v \right)$ is linearly dependent.

Can someone please guide me in through the proofs?

Answer 1

1. This has nothing to do with linear independence. If $\lambda_1u_1+\cdots+\lambda_mu_m+\mu v=0$ and $\mu\neq0$, then$$v=-\frac1\mu\left(\lambda_1u_1+\cdots+\lambda_mu_m\right)\in U.$$

2. If the family $u_1,\ldots,u_m$ is linearly independent and if $\lambda_1u_1+\cdots+\lambda_mu_m+\mu v=0$, then $\mu=0$. But then we have $\lambda_1u_1+\cdots+\lambda_mu_m=0$ and it follows from the linear independence of $u_1,\ldots,u_m$ that the $\lambda_k$'s are all equal to $0$. On the other hand if the family $u_1,\ldots,u_m$ is linearly dependent, then any larger family is also linearly dependent.

Answer 2

Assume $\mu\neq 0$ then $$v=-\frac{\lambda_1}{\mu}u_1-...-\frac{\lambda_m}{\mu}u_m\in U$$ contradiction. Can You take it from here?