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
- 1289 views
- $10.00
Related Questions
- Show that $tr(\sqrt{\sqrt A B \sqrt A})\leq 1$ , where both $A$ and $B$ are positive semidefinite with $tr(A)=tr(B)=1.$
- Decide if the following representations are linear representations.
- [Rotations in R^3 ] Consider R∶ R^3 → R^3 the linear transformation that rotates π/3 around the z-axis
- [Linear Algebra] $T$-invariant subspace
- Prove Property of Projection Matrices
- Get area of rotated polygon knowing all coordinates and angle.
- Find $x$ so that $\begin{pmatrix} 1 & 0 & c \\ 0 & a & -b \\ -\frac{1}{a} & x & x^2 \end{pmatrix}$ is invertible
- Linear Algebra - Vectors and Matrices
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