Space of all matrices with given column space
Let $V$ be the vector space of all $m \times n$ (assume $m \leq n$) matrices $M$ over a field $\mathbb{F}$ whose column space satisfies $\operatorname{colsp}(M) \subseteq W$, where $W \subseteq \mathbb{F}^m$ is a given subspace. Prove that $\operatorname{dim}_\mathbb{F}(V) = n \operatorname{dim}_\mathbb{F}(W)$.
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.
1 Attachment
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
- 724 views
- $10.00
Related Questions
- Show that eigenvectors of a symmetric matrix are orthogonal
- Linear independence of functions
- Linear algebra| finding a base
- Linear Algebra Question
- Diagonal and Similar Matrices
- Let $\mathbb{C} ^{2} $ a complex vector space over $\mathbb{C} $ . Find a complex subspace unidimensional $M$ $\subset \mathbb{C} ^{2} $ such that $\mathbb{C} ^{2} \cap M =\left \{ 0 \right \} $
- How to filter data with the appearance of a Sine wave to 'flattern' the peaks
- Find eigenvalues and eigenvectors of $\begin{pmatrix} 1 & 6 & 0 \\ 0& 2 & 1 \\ 0 & 1 & 2 \end{pmatrix} $
