The problem involves a subspace S of R3 spanned by two vectors, x and y. The task is to demonstrate that the orthogonal complement of S, denoted S⊥, is equal to the null space of a matrix A formed by these vectors.
Some participants have provided insights into the definitions and properties of the involved concepts. There is an ongoing exploration of the proof structure, with suggestions on how to demonstrate the equivalence of the two sets. No consensus has been reached yet, and multiple interpretations are being considered.
Participants are navigating potential misunderstandings regarding linear independence and the dimensionality of the subspace spanned by the vectors. There is also a note on the necessity of presenting the proof clearly and methodically.
Start with some definitions of the terms in this problem. Do you know what S[tex]\bot[/tex] and N(A) mean?IntroAnalysis said:Homework Statement
Let S be a subspace of R3 spanned by the vectors x = (x1, x2, x3)T and y = (y1, y2, y3)T
Let A = (x1 x2 x3 )
( y1 y2 y3)
Show that S[tex]\bot[/tex] = N(A).
Homework Equations
The Attempt at a Solution
Any hints?
Not necessarily. For example, the vectors <1, 1> and <2, 2>} span a subspace of R2 but they aren't linearly independent. This subspace has dimension 1.IntroAnalysis said:The subspace S spanned by the two vectors means that any vector in S can be written as a linear combination of x and y. This means x and y are linearly independent.
IntroAnalysis said:S[tex]\bot[/tex] is the orthogonal complement of S. It means the set w an element of R3 such that wTs= 0 for every s that's an element of S.
The null set N(A) is the set of all solutions to Ax = 0.
There exists z = (z1, z2, z3) orthogonal to x and y.
(z1, z2, z3)^T* (x1, x2, x3) = 0 ; and (z1, z2, z3)^T* (y1, y2, y3) = 0
and the nullspace of N(A) = (x1 x2 x3) * (z1, z2, z3)^T = x1z1 + x2z2 +x3z3 = 0
(y1, y2, y3) * (z1, z2, z3)^T = y1z1 + y2z2 + y3z3 = 0
Therefore, the orthogonal complement of S = N(A)
Does that work? Thanks for your assistance!