The discussion revolves around the orthogonal property of vectors in the context of vector spans, specifically addressing the relationship between a vector that is orthogonal to two other vectors and its orthogonality to their span.
Some participants have provided insights into the mathematical reasoning required to demonstrate the orthogonality of w to the span of u and v. There is an ongoing exploration of how to formally express this relationship using the properties of dot products and scalar multiplication.
Participants note the need to show that a vector in the span can be expressed in terms of scalars and that the original poster is seeking clarity on how to formally prove the orthogonality condition without assuming prior knowledge of the properties involved.
No, Grief said a vector in the subspace span(u,v) is of the form au+bv for scalars a and b. To say that a vector is orthogonal to a subspace means that the vector is orthogonal to each vector in that subspace.war485 said:you said that span(u,v) is in this form au+bv
You need to show that given a vector x in span(u,v), we have w.x=0 . From above, x is of the form au+bv, so you want to show that w.(au+bv)=0. This means beginning with w.(au+bv) and showing it equals 0. As Grief already said, to do so just requires distributivity and recognising that w being orthogonal to u and to v means that w.u=0 and w.v=0.war485 said:so
w . (au+bv) = 0
w . au + w . bv = 0
where a and b are any scalar numbers
and that's all? There's no more to it?