Question

in lecture 11 Gilbert strang uses an example of a vectorspace in 4 dimensions, where the vectors (v) whose components add up to zero is a subspace. He then asks what dimension is this subspace ? The answer is 3 dimensions but i dont understand why. Can anyone clarify this for me?

Answer 1

Hi,in this example,the typical vector in R^4 has four components,and thoes vectors contents V1+V2+V3+V4=0,we called this subspace is S,so S= the nullspace([1,1,1,1]),we kown,the matrix [1,1,1,1] has one pivot and there free variables,so,dim(nullspace)=3.

Answer 2

Another way to think about it that might help is, how many vectors does it take to create a basis for the subspace? An obvious basis is: (1, -1, 0, 0), (1, 0, -1, 0), and (1, 0, 0, -1). Since there are 3 vectors in the basis, the dim is 3.