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Questions and Answers
What does the theorem (u+v) + w = u +(v+w) and u + v = v + u state?
What does the theorem (u+v) + w = u +(v+w) and u + v = v + u state?
- Vectors cannot be combined
- Vectors are only associative
- Vectors can be subtracted
- Vectors are commutative and associative (correct)
What is a linear combination?
What is a linear combination?
c1u1+c2u2+...+cMuM
Define Span in the context of vectors.
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 _____
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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