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
63
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
1.5K
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
- 390 views
- $10.00
Related Questions
- Eigenvalues and eigenvectors of $\begin{bmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{bmatrix} $
- Find $a,b,c$ so that $\begin{bmatrix} 0 & 1& 0 \\ 0 & 0 & 1\\ a & b & c \end{bmatrix} $ has the characteristic polynomial $-\lambda^3+4\lambda^2+5\lambda+6=0$
- two short Linear Algebra questions
- Prove that $V={(𝑥_1,𝑥_2,⋯,𝑥_n) \in ℝ^n ∣ 𝑥_1+𝑥_2+...+𝑥_{𝑛−1}−2𝑥_𝑛=0}\}$ is a subspace of $\R^n$.
- Find the general solution of the system of ODE $X'=\begin{bmatrix} 1 & 3 \\ -3 & 1 \end{bmatrix} X$
- Hello! I Would like a proof detailed of the following question.
- Linear Algebra Help : Consider Two Planes, P1 and P2
- linear algebra
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