Questions and Answers
A vector v belongs to the span of S if and only if v = c1u1 + c2u2 + · · · + ckuk for some scalars c1, c2,..., ck.
True
The span of {0} is {0}.
True
If A = [u1 u2...uk] and the matrix equation Ax = v is inconsistent, then v does not belong to the span of {u1, u2,..., uk}.
True
If A is an m × n matrix, then Ax = b is consistent for every b in Rm if and only if the rank of A is n.
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The span of S is Rn if and only if the rank of [u1 u2...uk] is n.
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Every finite subset of Rn is contained in its span.
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If S1 and S2 are finite subsets of Rn such that S1 is contained in Span S2, then Span S1 is contained in Span S2.
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If S1 and S2 are finite subsets of Rn having equal spans, then S1 = S2.
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If S1 and S2 are finite subsets of Rn having equal spans, then S1 and S2 contain the same number of vectors.
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The spans of S and S ∪ {v} are equal if and only if v is in S.
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The span of a set of two nonparallel vectors in R2 is R2.
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The span of any finite nonempty subset of Rn contains the zero vector.
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If v belongs to the span of S, so does cv for every scalar c.
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If u and v belong to the span of S, so does u + v.
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The span of {v} consists of every multiple of v.
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If S is a generating set for Rm that contains k vectors, then k ≥ m.
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If A is an m × n matrix whose reduced row echelon form contains no zero rows, then the columns of A form a generating set for Rm.
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If the columns of an n × n matrix A form a generating set for Rn, then the reduced row echelon form of A is In.
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If A is an m × n matrix such that Ax = b is inconsistent for some b in Rm, then rank A < m.
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If S1 is contained in a finite set S2 and S1 is a generating set for Rm, then S2 is also a generating set for Rm.
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Study Notes
The Span of a Set of Vectors
- A vector v belongs to the span of a set S = {u1, u2, ..., uk} if it can be expressed as a linear combination of the vectors in S using scalars c1, c2, ..., ck.
- The span of the zero vector set {0} is solely {0} itself.
- If the matrix equation Ax = b is inconsistent for A = [u1, u2, ..., uk], then v does not belong to the span of {u1, u2, ..., uk}.
- For a matrix A of size m × n, Ax = b is consistent for all b in Rm if and only if the rank of A equals n; however, this statement is false.
- The span of a subset S = {u1, u2, ..., uk} equals Rn if and only if the rank of the matrix formed by these vectors is n.
- Every finite subset of Rn is inherently contained within its span.
- If S1 is a subset within the span of S2, then the span of S1 is also contained within the span of S2.
- Two finite subsets S1 and S2 can have equal spans without being identical; therefore, equal spans do not imply S1 = S2.
- It is also false that if S1 and S2 have equal spans, they must contain the same number of vectors.
- The spans of a nonempty set of vectors S and the set S ∪ {v} are equivalent only if v is already included in S.
- In R2, the span of two nonparallel vectors intersects the entirety of R2.
- The span of any finite nonempty subset in Rn includes the zero vector, as linear combinations can yield the zero vector.
- If a vector v belongs to the span of a set S, then any scalar multiple cv is also included in the span of S.
- For any two vectors u and v in the span of S, their sum u + v also belongs to the span of S.
- The span of a single vector {v} encompasses all scalar multiples of v.
- A set of vectors containing k must have k ≥ m to serve as a generating set for Rm.
- If the reduced row echelon form of an m × n matrix A includes no zero rows, the columns of A can generate Rm.
- The columns of an n × n matrix form a generating set for Rn, resulting in the reduced row echelon form being the identity matrix In.
- If the matrix equation Ax = b is inconsistent for some b in Rm, it implies that the rank of A is less than m.
- If S1 is contained within a finite set S2 and S1 can generate Rm, then S2 must also be capable of generating Rm.
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Description
This quiz covers the concept of the span of a set of vectors in linear algebra. It includes definitions and properties regarding vector spans and their implications for vector representation. Test your understanding of these fundamental concepts through flashcards.
