Homework Statement
Fix a real number a>1. If r=p/q is a rational number, we define a^r to be a^(p/q). Assume the fact that f(r)=a^r is a continuous increasing function on the domain Q of rational numbers r.
Let s be a real number. Prove that lim r--->s f(r) exists
Homework Equations...
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...
<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...