In the context of linear algebra, specifically regarding an $m \times n$ matrix $A$ (with $m$ rows and $n$ columns), the concepts of one-to-one (injective) and onto (surjective) are important when analyzing the linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(\mathbf{x}) = A\mathbf{x}$.

One-to-One (Injective):

A linear transformation $T$ is one-to-one if different inputs in the domain $\mathbb{R}^n$ map to different outputs in the codomain $\mathbb{R}^m$. In other words, if $T(\mathbf{u}) = T(\mathbf{v})$ implies $\mathbf{u} = \mathbf{v}$.

For an $m \times n$ matrix $A$:

• $T$ is one-to-one if the equation $A\mathbf{x} = \mathbf{0}$ has only the trivial solution $\mathbf{x} = \mathbf{0}$.
• This implies that the columns of $A$ are linearly independent.
• This condition is satisfied if and only if $\text{rank}(A) = n$. Thus, $n$ must be less than or equal to $m$ ($n \leq m$) for $T$ to be injective.

Onto (Surjective):

A linear transformation $T$ is onto if every vector in the codomain $\mathbb{R}^m$ is an image of at least one vector in the domain $\mathbb{R}^n$. In other words, for every $\mathbf{b} \in \mathbb{R}^m$, there exists an $\mathbf{x} \in \mathbb{R}^n$ such that $T(\mathbf{x}) = A\mathbf{x} = \mathbf{b}$.

For an $m \times n$ matrix $A$:

• $T$ is onto if the matrix equation $A\mathbf{x} = \mathbf{b}$ has at least one solution for every $\mathbf{b} \in \mathbb{R}^m$.
• This implies that the columns of $A$ span $\mathbb{R}^m$.
• This condition is satisfied if and only if $\text{rank}(A) = m$. Thus, $m$ must be less than or equal to $n$ ($m \leq n$) for $T$ to be surjective.

Key Rules:

1. One-to-One Condition: $\text{rank}(A) = n$. This requires $n \leq m$.
2. Onto Condition: $\text{rank}(A) = m$. This requires $m \leq n$.
3. If $A$ is a square matrix ($m = n$):
   • $A$ is one-to-one if and only if it is onto.
   • $A$ is invertible (nonsingular) if and only if $\text{rank}(A) = n$.

Understanding the Concepts:

• For a transformation to be one-to-one, no two distinct vectors in the domain should map to the same vector in the codomain.
• For a transformation to be onto, every vector in the codomain must be reachable by applying the transformation to some vector in the domain.

Relative Questions:

1. How does the concept of the null space relate to a matrix being one-to-one?
2. What is the relationship between the row space and a matrix being onto?
3. How do the dimensions of the domain and codomain affect whether a linear transformation can be both one-to-one and onto?
4. How can you determine if a given $m \times n$ matrix is invertible?
5. What happens to the properties of one-to-one and onto if the matrix $A$ is not of full rank?

Tip:

To quickly check if an $m \times n$ matrix is one-to-one or onto, compute its rank and compare it to $m$ and $n$. The rank provides crucial information about the linear independence of columns and the span of rows.