# 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.

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?

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.
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?