Two Abstract Algebra Questions

1. Let V be a vector space over a field F, and $\displaystyle v_1, v_2,...,v_n \in V$. If there is an i>1 such that $\displaystyle v_i \in span(v_1, v_2,...,v_{i-1})$, then $\displaystyle span(v_1,...,v_n)=span(v_1,...,v_{i-1},v_{i+1},...,v_n)$.

I'm not sure how to start this problem. I don't know what properties to apply to this problem.

2. Let V be a vector space over a field F, and $\displaystyle v_1, v_2,...,v_n \in V$. Let $\displaystyle u_i = v_i-a_i v_1$, where $\displaystyle a_i \in F$ and $\displaystyle 2\leq i \leq n$. Show that if $\displaystyle u_2,...,u_n$ are linearly dependent, then $\displaystyle v_1,...,v_n$ are also linearly dependent.

Quote:

---

Originally Posted by grad444

1. Let V be a vector space over a field F, and $\displaystyle v_1, v_2,...,v_n \in V$. If there is an i>1 such that $\displaystyle v_i \in span(v_1, v_2,...,v_{i-1})$, then $\displaystyle span(v_1,...,v_n)=span(v_1,...,v_{i-1},v_{i+1},...,v_n)$.

I'm not sure how to start this problem. I don't know what properties to apply to this problem.

---

you need to show that $\displaystyle span(v_1,...,v_n) \subseteq span(v_1,...,v_{i-1},v_{i+1},...,v_n)$ and $\displaystyle span(v_1,...,v_{i-1},v_{i+1},...,v_n) \subseteq span(v_1,...,v_n).$ to prove that $\displaystyle span(A) \subseteq span(B),$ you only need to

show that $\displaystyle A \subseteq span(B).$

Quote:

---

2. Let V be a vector space over a field F, and $\displaystyle v_1, v_2,...,v_n \in V$. Let $\displaystyle u_i = v_i-a_i v_1$, where $\displaystyle a_i \in F$ and $\displaystyle 2\leq i \leq n$. Show that if $\displaystyle u_2,...,u_n$ are linearly dependent, then $\displaystyle v_1,...,v_n$

are also linearly dependent.

---

$\displaystyle \sum_{i=2}^n c_i u_i =\sum_{i=1}^n c_i v_i,$ where $\displaystyle c_1=-\sum_{i=2}^nc_ia_i.$ now if $\displaystyle u_2, \cdots , u_n$ are linearly dependent, then there exist $\displaystyle c_2, \cdots , c_n,$ not all 0, such that $\displaystyle \sum_{i=2}^n c_i u_i =0.$ thus ...