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3 cards from this section. Quick, good the night before a test.
Define what a basis is in the context of vector spaces.
Consider how vectors relate to each other in terms of dependencies.
What is the dimension of R⁵?
Think about how many independent directions are in the space.
What is a basis of a vector space?
Think about the criteria a set must meet to be considered a basis.
Does the zero vector space have a basis?
Consider how many independent vectors are necessary for a basis.
Given a vector space consisting of vectors in R³, determine if the set {(1,0,0), (0,1,0), (0,0,1), (1,1,1)} is a basis. Explain your reasoning.
Check if the last vector can be formed using coefficients from the first three.
If a vector space V has dimension n, show that no more than n vectors can be linearly independent in V.
Consider how adding vectors beyond the dimension interacts with independence.
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