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
- 1248 views
- $10.00
Related Questions
- Frontal solver by Bruce Irons? Am I using the right Algorithm here?
- [Linear Algebra] Proof check. Nilpotent$\Rightarrow Spec\Rightarrow$ Characteristic Polynomial $\Rightarrow$ Nilpotent
- The Span and Uniqueness of Solutions in a Parametric Matrix
- Linear Algebra - Matrices (Multiple Choice Question) (1st Year College)
- Length of a matrix module
- Show that eigenvectors of a symmetric matrix are orthogonal
- Linear Algebra Question
- Hamming metric isometries
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