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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 (A)
The span of {0} is {0}.
The span of {0} is {0}.
True (A)
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 (A)
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.
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.
Every finite subset of R^n is contained in its span.
Every finite subset of R^n is contained in its span.
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.
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.
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.
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.
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.
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.
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.
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.
The span of {v} consists of every multiple of v.
The span of {v} consists of every multiple of v.
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.
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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.
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