This document contains a summary of key concepts from a linear algebra course. It discusses vectors and vector spaces. Specifically, it examines whether a set of vectors forms a subspace and finds a basis for a subspace. It shows that a set V is a subspace of R4 by proving it is closed under addition and scalar multiplication. It then determines a basis B of three vectors for the subspace V, showing its dimension is 3.

1. SIT292 LINEAR ALGEBRA 2013
Vectors and spaces: Linear algebra
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Abstract
a)
Let w=(-1,1,1,1)
non-empty.
Let v1,
Now:
(v1+v2).w = v1.w +v2.w = 0 + 0 = 0
and
(cv1).w = c(v1.w) = c0 =0
Hence V is a subspace of R4.
b)
if xT(-1,1,1,1)=0
-x1+x2+x3+x4=0
=>
x4=x1-x2-x3
Hence we can write it as follows:
x=x1(1,0,0,1) + x2(0,1,0,-1) + x3(0,0,1,-1)
where xT is x transpose.
Thus B={(1,0,0,1),(0,1,0,-1),(0,0,1,-1)} is a basis of V. The dimension of V is 3.