Questions and Answers
What does the theorem (u+v) + w = u +(v+w) and u + v = v + u state?
What is a linear combination?
c1u1+c2u2+...+cMuM
Define Span in the context of vectors.
span(u1,u2,...uM) = {łu1, łu2..., łuM FOR ł1, ł2,... łM exists in Rn}
The theorem states that u1, u2,... uM exists in Rn where M is _____
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Study Notes
Vector Theorems
- Theorem states that for any vectors u, v, and w, addition is both commutative and associative:
- (u + v) + w = u + (v + w)
- u + v = v + u
Linear Combination
- A linear combination involves scalars c1, c2,..., cM from the real numbers (R) and vectors u1, u2,..., uM from Rⁿ.
- It is expressed as:
- c1u1 + c2u2 + ... + cMuM
Span
- Given vectors u1, u2,..., uM in Rⁿ, their span includes all possible linear combinations formed by these vectors.
- Mathematically represented as:
- span(u1, u2,..., uM) = {λ1u1, λ2u2,..., λMuM | λ1, λ2,..., λM ∈ R}
General Theorem for Vectors
- A general theorem applies to vectors u1, u2,..., uM existing in Rⁿ, concerning their properties and relationships within a vector space.
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Description
Test your knowledge of key concepts in linear algebra with these flashcards from UNC. Learn about vector addition, linear combinations, and the span of vectors. Perfect for students looking to reinforce their understanding of linear algebra principles.
