Understanding Notation and Proving Linear Independence

[cateater2000]

Hi I just need some help on understanding some general notation in this quesiton:

Prove if {x_1,x_2,..,x_m} is linearly independent then so is {x_1,x_2,...,x_i-1, x_i+1,...,x_m} for every i in {1,2,...,m}.


I don't really understand what the difference between {x_1,x_2,...,x_i-1, x_i+1,...,x_m} for every i in {1,2,...,m} and {x_1,x_2,..,x_m} is.

Any help clarifying this would be great, and any hints for the question would be must appreciated, thanks.

 

[matt grime]

The second set omits the i'th vector.

Eg the large set is {a,b,c,d} and there are 4 other sets: {b,c,d}, {a,c,d}, {a,b,d}, {a,b,c}

 

[TD]

If you understand that, you can easily prove this by contradiction.
Suppose one of the smaller sets is linearly dependent, then one of its elements is a lineair combination of the others. What does that tell you about the larger set then?