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One-to-One & Onto
Linear Transformations

Understand when a function maps distinct inputs to distinct outputs and when it covers every element of the codomain: the two pillars of invertibility.

Linear Algebra

Quick Refresher: Functions

A function \( f : A \to B \) assigns to every element of the domain \( A \) exactly one element of the codomain \( B \). The set of all actual outputs is the range (or image), which may be smaller than \( B \).

Domain A Codomain B 1 2 3 a b c f

What Does One-to-One Mean?

A function \( f : A \to B \) is one-to-one (injective) if different inputs always produce different outputs:

A B 1 2 3 a b c d ✓ One-to-One A 1 2 3 ✕ Not One-to-One a b

On the right, inputs 1 and 2 both map to a: that collision violates injectivity.

What Does Onto Mean?

A function \( f : A \to B \) is onto (surjective) if every element of \( B \) is hit by at least one element of \( A \):

A B 1 2 3 4 a b c ✓ Onto A 1 2 a b c (missed) ✕ Not Onto

On the right, element c in \( B \) has no arrow pointing to it: the function fails to be surjective.

Why Do These Matter?

One-to-One
Guarantees no information is lost: you can undo the mapping because each output came from a unique input.
Onto
Guarantees every target is reachable: the function covers the entire codomain, leaving no gaps.
Both (Bijection)
When a function is both one-to-one and onto, it has an inverse: \( f^{-1} \) exists and is itself a function.

The Matrix Conditions

Let \( T(\mathbf{x}) = A\mathbf{x} \) where \( A \) is an \( m \times n \) matrix. The properties of \( T \) are encoded in \( A \):

1
One-to-One \(\iff\) the columns of \( A \) are linearly independent \(\iff\) \( A\mathbf{x} = \mathbf{0} \) has only the trivial solution \(\iff\) every column has a pivot.
2
Onto \(\iff\) the columns of \( A \) span \( \mathbb{R}^m \) \(\iff\) \( A\mathbf{x} = \mathbf{b} \) is consistent for every \( \mathbf{b} \) \(\iff\) every row has a pivot.

Reading Pivots from Row Echelon Form

3 pivots (\(3 \times 3\))
1 2 3 0 1 5 0 0 1 ✓ One-to-One & Onto
2 pivots (\(2 \times 3\))
1 3 2 0 1 4 ✕ Not One-to-One (free variable) ✓ Onto (pivot in every row)
For an \(m \times n\) matrix:
One-to-One requires \(n\) pivots (pivot in every column)
Onto requires \(m\) pivots (pivot in every row)
Both require \(m = n\) with \(n\) pivots

One-to-One vs. Onto: Side by Side

One-to-One (Injective) Onto (Surjective)
Question asked Do different inputs always give different outputs? Is every output in \( B \) actually reached?
Formal test \( f(x_1)=f(x_2) \Rightarrow x_1=x_2 \) \( \forall\, b \in B,\; \exists\, a: f(a)=b \)
Matrix test Pivot in every column Pivot in every row
Dimension need \( m \geq n \) \( n \geq m \)
Kernel test \( \ker(T) = \{\mathbf{0}\} \) \( \text{range}(T) = \mathbb{R}^m \)

The Dimension Constraint

For \( T : \mathbb{R}^n \to \mathbb{R}^m \) (an \( m \times n \) matrix), the number of pivots is at most \( \min(m,n) \). This forces constraints:

\( n > m \)
More columns than rows: cannot be one-to-one (free variables must exist). Could be onto.
\( m > n \)
More rows than columns: cannot be onto (not enough pivots for all rows). Could be one-to-one.
\( m = n \)
Square matrix: one-to-one \(\iff\) onto. If one holds, so does the other. Invertible!

How to Check the Conditions

1
Write the matrix \( A \) whose columns are the images of the standard basis vectors (or given directly).
2
Row reduce \( A \) to echelon form. Count the pivot positions.
3
Check columns: if every column has a pivot → one-to-one.
4
Check rows: if every row has a pivot → onto.
5
Conclude: both → bijection (invertible). Neither → neither. One without the other → partial property only.

Example 1: A Square Matrix

Let \( A = \begin{bmatrix} 1 & 2 \\ 3 & 7 \end{bmatrix} \). Is \( T(\mathbf{x}) = A\mathbf{x} \) one-to-one? Onto?

Example 2: More Columns than Rows

Let \( A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \end{bmatrix} \). This maps \( \mathbb{R}^3 \to \mathbb{R}^2 \).

✕ Not One-to-One
Column 3 is free: \( A\mathbf{x} = \mathbf{0} \) has infinitely many solutions. For example, \( \mathbf{x} = \begin{bmatrix} -2 \\ -3 \\ 1 \end{bmatrix} \).
✓ Onto
Every row has a pivot (2 pivots, 2 rows), so the columns span all of \( \mathbb{R}^2 \).

Example 3: More Rows than Columns

Let \( A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \\ 0 & 0 \end{bmatrix} \). This maps \( \mathbb{R}^2 \to \mathbb{R}^3 \).

✓ One-to-One
Every column has a pivot (2 pivots, 2 columns). The only solution to \( A\mathbf{x} = \mathbf{0} \) is \( \mathbf{x} = \mathbf{0} \).
✕ Not Onto
Row 3 has no pivot: \( A\mathbf{x} = \begin{bmatrix} 0\\0\\1 \end{bmatrix} \) is inconsistent. Not every \( \mathbf{b} \in \mathbb{R}^3 \) is reachable.

Connection to Kernel and Range

Domain ker(T) Codomain range(T) T
One-to-One \(\iff \ker(T) = \{\mathbf{0}\}\)
Onto \(\iff \text{range}(T) = \text{Codomain}\)

Common Mistakes to Avoid

✕ Confusing Row and Column
Pivot in every column → one-to-one. Pivot in every row → onto. Mixing these up reverses the conclusion.
✕ Assuming Square
For non-square matrices, one-to-one does NOT imply onto (and vice versa). This equivalence only holds for square matrices.
✕ Ignoring Dimensions
A map from \(\mathbb{R}^4 \to \mathbb{R}^2\) can never be one-to-one. Always check \(m\) vs. \(n\) before row reducing.

Question 1 of 5

A linear transformation \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) has the matrix \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \). Which statement is true?

Question 2 of 5

If an \( m \times n \) matrix \( A \) has \( n > m \), which is impossible for \( T(\mathbf{x}) = A\mathbf{x} \)?

Question 3 of 5

The matrix \( A = \begin{bmatrix} 1 & 3 \\ 2 & 6 \end{bmatrix} \) row reduces to \( \begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix} \). What can we conclude?

Question 4 of 5

For a square matrix \( A \), which statement is equivalent to "\( T \) is one-to-one"?

Question 5 of 5

A \( 3 \times 2 \) matrix \( A \) has pivot positions in both columns. Which is true?