Some technical questions about spans and bases

by Bipolarity
Tags: bases, spans, technical
Bipolarity is offline
Jan6-13, 04:14 PM
P: 783
I have some technical questions about spans and bases which my textbook really does not cover very well. I would appreciate answers to them. They are not textbook problems, merely specific quesitons relating to the definitions of "span" and "basis".

1) If span(S) = V, need the elements of S be in V?
2) If S is a basis for V, need the elements of S be in V?
3) If S spans V, does S span every subspace of V?
4) If S is a basis for V, is S a basis for every subspace of V?
5) If S is linearly independent, is every subset of S also linearly independent?
6) If S is linearly dependent, is every superset of S also linearly dependent?

If I am not wrong, the answers to the questions are:
1) Yes
2) Yes
3) Yes
4) No
5) Yes
6) Yes

But I would like explicit confirmation.
All help is appreciated. Thanks.

Phys.Org News Partner Science news on
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
mfb is offline
Jan6-13, 04:37 PM
P: 10,767
All correct. For a strict mathematical formulation, you should let everything be a part of a vector space W (can be identical to V, but then questions 1 and 2 are meaningless).

3) If S spans V, does S span every subspace of V?
The span of S contains every subspace of V as subset.
AlephZero is offline
Jan6-13, 06:16 PM
Sci Advisor
HW Helper
P: 6,341
I disagree about (3).

If S spans V and W is a subspace of V, then the elements of S may not all be in W, so "S spans W" might contradicts the (correct IMO) answer to (1).

Also the definition of "span" in implies that there is only ONE space that is spanned by a given set S - namely, the intersection of all subspaces of V that contain (all the vectors in) S.

Register to reply

Related Discussions
Please help me by asking technical questions! Career Guidance 2
Topology questions (bases) Topology and Analysis 14
[Chemistry] questions on solubilty of bases and naphthalen-2-ol Biology, Chemistry & Other Homework 1
Bases and Dimension questions Calculus & Beyond Homework 3
Matrices: Technical Questions Linear & Abstract Algebra 1