Question

Suppose v1;v2;vm is linearly independent in V and w 2 V. Prove that dim span (v1 +w;v2 +w;:::;vm +w) m1.

Answer 1

If there are (m+1) linearly independent vectors then dim $V\ge m+1>m ⇒⇐.$ This is the contrapositive. Let m be dimension of V and V_i E V for I=1 to m+1 be set of linearly independent set.

Hence V_i=I to m are also linearly independent this V_i for I=1 to m is basis for V. We need to prove that. Let let W=span {V_1,V_2......V_m}, then W$\le$ V and dim (W)=dim(V)=m, then W=V.

Since V_m+1 E V, has a unique representation using basis vectors, thus

$V_{m+1}=\displaystyle\sum_{i=1}^m \alpha _i \space V_i$

Hence V_i,i E 1. ...,m+1 are linearly dependent.