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 x n matrix, then Ax = b is consistent for every b in R^m if and only if the rank of A is n.
If A is an m x n matrix, then Ax = b is consistent for every b in R^m if and only if the rank of A is n.
Signup and view all the answers
The span of a set S is R^n if and only if the rank of [u1 u2...uk] is n.
The span of a set S is R^n if and only if the rank of [u1 u2...uk] is n.
Signup and view all the answers
Every finite subset of R^n is contained in its span.
Every finite subset of R^n is contained in its span.
Signup and view all the answers
If S_1 and S_2 are finite subsets of R^n such that S_1 is contained in Span S_2, then Span S_1 is contained in Span S_2.
If S_1 and S_2 are finite subsets of R^n such that S_1 is contained in Span S_2, then Span S_1 is contained in Span S_2.
Signup and view all the answers
If S_1 and S_2 are finite subsets of R^n having equal spans, then S_1 = S_2.
If S_1 and S_2 are finite subsets of R^n having equal spans, then S_1 = S_2.
Signup and view all the answers
If S_1 and S_2 are finite subsets of R^n having equal spans, then S_1 and S_2 contain the same number of vectors.
If S_1 and S_2 are finite subsets of R^n having equal spans, then S_1 and S_2 contain the same number of vectors.
Signup and view all the answers
Let S be a nonempty set of vectors in R^n, and let v be in R^n. The spans of S and S U {v} are equal if and only if v is in S.
Let S be a nonempty set of vectors in R^n, and let v be in R^n. The spans of S and S U {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 R^2 is R^2.
The span of a set of two nonparallel vectors in R^2 is R^2.
Signup and view all the answers
The span of any finite nonempty subset of R^n contains the zero vector.
The span of any finite nonempty subset of R^n 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 R^m that contains k vectors, then k >= m.
If S is a generating set for R^m that contains k vectors, then k >= m.
Signup and view all the answers
Study Notes
The Span of a Set of Vectors
- The span of a set of vectors S = {u1, u2,..., uk} in R^n includes all linear combinations of those vectors, i.e., v = c1u1 + c2u2 +...+ ckuk for scalars c1, c2,..., ck.
- The span of the set containing only the zero vector, {0}, is {0}, as the only linear combination results in the zero vector.
Matrix Inconsistency and Spans
- Inconsistent matrix equation Ax = v indicates that vector v is not part of the span of {u1, u2,..., uk} since it cannot be expressed as a linear combination of those vectors.
- An m x n matrix A guarantees a consistent equation Ax = b for any b in R^m if the rank of A is m, not n.
Rank and Spanning Sets
- The span of S is equal to R^n if and only if the rank of the matrix formed by the vectors in S is n.
- Every finite subset of R^n is included in its own span since spans are formed from linear combinations of their components.
Relationships Between Subsets and Their Spans
- If S_1 is contained in the span of S_2, then the span of S_1 must also be contained within the span of S_2.
- Two finite subsets S_1 and S_2 can have equal spans without being identical; example: S_1 = {[1;0], [2;0]} and S_2 = {[1;0]} have equal spans but different cardinalities.
- For two finite subsets S_1 and S_2 with equal spans, they do not necessarily contain the same number of vectors.
Inclusion of Vectors in Spans
- A nonempty set S will have equivalent spans whether or not an additional vector v is included, provided that v is in the span rather than in S.
- In R^2, the span of two nonparallel vectors spans the whole space R^2, as they can generate any vector in that space through linear combinations.
- The zero vector is always included in the span of any finite non-empty subset of R^n, as it can be generated by multiplying any vector by the scalar zero.
Scalar Multiplication and Addition within Spans
- If a vector v belongs to the span of S, then any scalar multiple cv also belongs to the span.
- Adding two vectors u and v from the span of S results in another vector u + v that is also within the span due to closure under addition.
Properties of Single-Vector Spans
- The span of a single vector {v} comprises all scalar multiples of v, capturing all linear combinations of v.
Conditions for Generating Sets
- A generating set S for R^m, which contains k vectors, must have k greater than or equal to m to ensure that the rank of the corresponding matrix equals m.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of the concept of the span of a set of vectors in Linear Algebra with this True/False quiz. Each statement challenges your knowledge on the properties and definitions associated with spans. Ideal for students looking to reinforce their understanding of vector spaces.
