Vector space axioms, span, basis, and dimension from a college linear algebra course.
29 cards · basic cards · AI-written, checked twice. Edit anything.
- What are the 10 axioms that define a vector space?
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Closure under addition, closure under scalar multiplication, associativity of addition, commutativity of addition, additive identity (zero vector), additive inverses, associativity of scalar multiplication, distributivity of scalar multiplication over vector addition, distributivity of scalar multiplication over field addition, scalar multiplication identity.- If u and v are in vector space V, what must be true about u + v?
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u + v must also be in V (closure under addition).- If v is in vector space V and c is a scalar, what must be true about cv?
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cv must also be in V (closure under scalar multiplication).- Define a subspace.
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A subset W of a vector space V that is itself a vector space under the same operations.- State the subspace test (two-condition version).
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A nonempty subset W of vector space V is a subspace if and only if: (1) W is closed under addition, and (2) W is closed under scalar multiplication.- Must the zero vector be in every subspace?
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Yes, every subspace must contain the zero vector.- Define the span of a set of vectors.
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The set of all linear combinations of those vectors.- Is the span of a set of vectors always a subspace?
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Yes, the span of any set of vectors is a subspace of the containing vector space.- What does it mean for a set of vectors to be linearly independent?
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The only solution to c1*v1 + c2*v2 + ... + cn*vn = 0 is c1 = c2 = ... = cn = 0.- What does it mean for a set of vectors to be linearly dependent?
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There exist scalars c1, c2, ..., cn, not all zero, such that c1*v1 + c2*v2 + ... + cn*vn = 0.- When is a set of vectors trivially linearly dependent?
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When it contains the zero vector.- Define a basis for a vector space.
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A set of vectors that is linearly independent and spans the vector space.- What is the dimension of a vector space?
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The number of vectors in a basis for that vector space.- Are all bases for a finite-dimensional vector space the same size?
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Yes, all bases have the same number of vectors.- What is the dimension of R^n?
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n.
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