Space of matrices with bounded row space
Let $V$ be a vector space of $m \times n$ matrices over a field $\mathbb{F}$ with the property that, for all $M \in V, \ \operatorname{rowsp}(M) \subseteq W$, where $W \subseteq \mathbb{F}^n$ is a subspace which is a row space of some $M \in V$. Prove that $\operatorname{dim}_\mathbb{F}(V) = \operatorname{max}\{n,m\} \operatorname{dim}_\mathbb{F}(W)$.
Ribs
114
Answer
Answers can only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
Martin
1.7K
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 1198 views
- $10.00
Related Questions
- Find where this discrete 3D spiral converges in explict terms
- Linear Algebra Exam
- How do I evaluate and interpret these sets of vectors and their geometric descriptions?
- Show that eigenvectors of a symmetric matrix are orthogonal
- Length of a matrix module
- [Linear Algebra] Proof check. Nilpotent$\Rightarrow Spec\Rightarrow$ Characteristic Polynomial $\Rightarrow$ Nilpotent
- [ eigenvalues and eigenvectors] Prove that (v1, v2, v3) is a basis of R^3
- Consider the plane in R^4 , calculate an orthonormal basis
If you want equality for the dimension you need to also assume that W is itself the row space of some M in V. Otherwise V=0 is a counterexample.
You're right, edited, thanks