Podcast
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.
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}.
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}.
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.
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.
Signup and view all the answers
The span of S is Rn if and only if the rank of [u1 u2...uk] is n.
The span of S is Rn if and only if the rank of [u1 u2...uk] is n.
Signup and view all the answers
Every finite subset of Rn is contained in its span.
Every finite subset of Rn is contained in its span.
Signup and view all the answers
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.
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.
Signup and view all the answers
If S1 and S2 are finite subsets of Rn having equal spans, then S1 = S2.
If S1 and S2 are finite subsets of Rn having equal spans, then S1 = S2.
Signup and view all the answers
If S1 and S2 are finite subsets of Rn having equal spans, then S1 and S2 contain the same number of vectors.
If S1 and S2 are finite subsets of Rn having equal spans, then S1 and S2 contain the same number of vectors.
Signup and view all the answers
The spans of S and S ∪ {v} are equal if and only if v is in S.
The spans of S and S ∪ {v} are equal if and only if v is in S.
Signup and view all the answers
The span of a set of two nonparallel vectors in R2 is R2.
The span of a set of two nonparallel vectors in R2 is R2.
Signup and view all the answers
The span of any finite nonempty subset of Rn contains the zero vector.
The span of any finite nonempty subset of Rn contains the zero vector.
Signup and view all the answers
If v belongs to the span of S, so does cv for every scalar c.
If v belongs to the span of S, so does cv for every scalar c.
Signup and view all the answers
If u and v belong to the span of S, so does u + v.
If u and v belong to the span of S, so does u + v.
Signup and view all the answers
The span of {v} consists of every multiple of v.
The span of {v} consists of every multiple of v.
Signup and view all the answers
If S is a generating set for Rm that contains k vectors, then k ≥ m.
If S is a generating set for Rm that contains k vectors, then k ≥ m.
Signup and view all the answers
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.
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.
Signup and view all the answers
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.
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.
Signup and view all the answers
If A is an m × n matrix such that Ax = b is inconsistent for some b in Rm, then rank A < m.
If A is an m × n matrix such that Ax = b is inconsistent for some b in Rm, then rank A < m.
Signup and view all the answers
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.
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.
Signup and view all the answers
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
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.
