Recent content by Krovski

Homework Statement I'm using Introduction to Analysis 5th edition by Edward D. Gaughan. The question is: Prove (De Morgan) S\(\bigcap A_{\lambda}) = \cup(S\A) \lambda\epsilon \Lambda Where \Lambda A and S are sets (doesn't specify real or complex but assuming real) Homework...

From what I understand, a basis is essentially a subset of a vector space over a given field. Now what I'm not so sure of is the linearly independence part. If the basis has two linearly independent vectors, then than means they aren't collinear: rather, they wouldn't have the same slope...

Yes I have been trying to read this one text, but I'm definitely behind the power curve for this topic since I haven't had much experience or luck doing proofs. I have to think of everything graphically which can become hard to do when dealing in higher dimensions. So, what I have seen...

to clarify, W=(a1,a2,a3)\epsilon ℝ^3 (this set is a subspace in ℝ^3) so would W1=(a2+2,a2,a3) also be a subsace in ℝ^3? or does that violate the definition of a subspace with the a2+2 term?

ok thank you. more precisely, if W=(a1,a2,a3) would W be a subspace of \textbf{R^3} if, say, we had one element a1=a2+2? then W=(a2+2,a2,a3) it technically isn't seen as c(a1,a2,a3) or even adding another vector in \textbf{R^3} so that this vector, we'll call, (b1,b2,b3) means...

this is exactly why I need help understanding what a subspace is lol. so would i then say I need to show that W is also a vector space? and that it all its elements also belong in V? or am I thinking of subsets?

I'm having trouble conceptualizing exactly what a subspace is and how to identify subspaces from vector spaces. I know that the definition of a subspace is: A subset W of a vector space V over a field \textbf{F} is a subspace if W is also a vector space over \textbf{F} w/ the operations of...

good catch and thank you

<v,v>=0 only when v=0 the definition for inner-product is Ʃ(v_{j})(\overline{v_{j}}) for 1≤j≤n where n is the length of vector v note that \overline{v_{j}} is defined as the adjoint, or conjugate transpose when dealing in ℝ, you'll never get <v,v>=0 because it is merely taking the...

So does that mean that every non-zero finite-dimensional inner-product space has an orthonormal basis? would that basis then be unique?