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 $\operatorname{dim}_\mathbb{F}(V) = n \operatorname{dim}_\mathbb{F}(W)$.
107
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
1.6K
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
- 516 views
- $10.00
Related Questions
- Linear Algebra Assistance: Linear Combinations of Vectors
- Find $a,b,c$ so that $\begin{bmatrix} 0 & 1& 0 \\ 0 & 0 & 1\\ a & b & c \end{bmatrix} $ has the characteristic polynomial $-\lambda^3+4\lambda^2+5\lambda+6=0$
- Linearly independent vector subsets.
- Frontal solver by Bruce Irons? Am I using the right Algorithm here?
- Diagonalization of linear transformations
- Relating dot product divided with square of the vector while changing basis of vector
- Numerical Linear Algebra Question
- Find the eigenvalues of $\begin{pmatrix} -1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 3 & -1 \end{pmatrix} $