# Definition

The **rank **of a matrix is the number of non-zero rows in its RREF.

# Injective

## Theorem: Injective means Tx = 0 is trivial

## Theorem: Ax = 0 is trivial means columns of A are linearly independent

## Theorem: If the columns of A are L.I., then the number of pivots equals n.

# Surjective

## Theorem: A is surjective means that there m pivots.

We conclude that A has m pivots.

# Row Space

A row space is the span of the rows of a matrix. The row space of two row-equivalent matrices is the same. Why? because to row reduce you just add linear combinations of rows. The basis of the row space is equal to the non-zero rows in the row-reduced matrix, which equals the dimension of the row space.

# Column Space

The column space is the span of the columns of A. In other words, it’s the image of a matrix A. I will prove that the dimension of the column space equals the dimension of the row space which equals the rank.

# Null Space

The null space is the set of all x such that Ax = 0. I will later prove that the Rank + Dim (Null Space) = n (columns in A).