Vector spaces Base and its subspace Basis

Question

I have a vector space $V$ and a subspace as described.

$$V = F^n, W = \{(a_1,...,a_n) | a_1 + ... + a_n = 0\}$$ I need to find a Basis for $W$ and than expand it to be Basis for $V$ thanks for the help.

Answer 1

At first observe that, solving $a_1 + \cdots + a_n = 0$, we obtain:
$$a_n = t_n, a_{n-1} = t_{n-1}, \cdots , a_2 = t_2, a_1 =-t_n-\cdots-t_2$$ All the vectors of W are of the form: $$\begin{pmatrix} -t_n \cdots -t_2 \\ -t_2 \\ \vdots \\ -t_n \end{pmatrix}$$ That can be written in the form: $$t_2\begin{pmatrix} -1\\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix} + t_3\begin{pmatrix} -1\\ 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix} +\cdots +t_n\begin{pmatrix} -1\\ 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix} $$ So all the vector of W can be written as a linear combination of $$\{\begin{pmatrix} -1\\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix},\begin{pmatrix} -1\\ 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix},\cdots,\begin{pmatrix} -1\\ 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix}\} $$ that is a base for W. Can you go ahead now?

Answer 2

Since you have $n$ varibles and only one equation, subspace W has dimension $n-1$.

You can find a basis for W by the $n-1$ conditions:

$$\begin{cases}a_1=-a_2=1, \quad a_3=a_4=...=a_n=0\\a_2=-a_3=1, \quad a_1=a_4=...=a_n=0\\...\\a_{n-1}=-a_n=1, \quad a_1=a_2=...=a_{n-2}=0\end{cases}$$

that is

$$\begin{cases}v_1=(1,-1,0,...,0,0)\\ v_2=(0,1,-1,...,0,0)\\ ...\\ v_{n-1}=(0,0,0,..,1,-1)\end{cases}$$

Then you can extend this basis to V looking for a vector wich do not satisfies the given conditions, e.g.:

$$a_i=1\neq0$$

that is

$$v_{n}=(1,1,1,..,1,1)$$