Row space

Dependencies:

  1. Matrix
  2. Vector Space
  3. Matrices over a field form a vector space
  4. /linear-algebra/vector-spaces/condition-for-subspace

Let $A$ be a matrix. The row space of $A$ is the set of all linear combinations of the rows of $A$.

The row space is a vector space.

Proof that row space is a vector space

Let $A$ be an $m$ by $n$ matrix. Let $V$ be the set of all linear combinations of rows of $A$. Then $V \subseteq \mathbb{R}^n$.

We know that $\mathbb{R}^n$ is a vector space. Since $V$ is closed under addition and scalar multiplication, $V$ is a subspace of $\mathbb{R}^n$. Hence, $V$ is a vector space.

Dependency for:

  1. Rank of a matrix
  2. Row equivalent matrices have the same row space

Info:

Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. Group
  3. Ring
  4. Field
  5. Vector Space
  6. Semiring
  7. Matrix
  8. Matrices over a field form a vector space