Matrices as Functions
Introduced| English | Meaning |
|---|---|
| input/ˈɪnpʊt/ | A value, signal or resource supplied to a function, program, device or process. |
| output/ˈaʊtpʊt/ | A result produced by a function, calculation, device or program. |
| composition/ˌkɒmpəˈzɪʃn/ | Combining functions by applying one function to the output of another. |
| inverse function/ɪnˈvɜːs ˈfʌŋkʃn/ | A function that reverses another function's mapping on a domain where that mapping is one-to-one. |
A matrix is a function
- You have used functions that take a number and return a number.
- A matrix takes a vector and returns a vector — it, too, is a function.
- Its "rule" is: multiply the input vector by the matrix.
- Seeing a matrix this way connects it to everything you know about functions.
Input in, output out
- The input 输入 is a vector; the matrix processes it; the output 输出 is a new vector.
- $\begin{bmatrix}2&0\\0&3\end{bmatrix}$ takes $\langle 4, 5\rangle$ to $\langle 8, 15\rangle$ — it stretches each direction.
- Think of the matrix as a machine: vector in the top, transformed vector out the bottom.
- A whole region of inputs maps to a whole region of outputs.
Viewing a 2 × 2 matrix as a function, its input and output are both…
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A matrix-as-function takes an input vector and returns an output vector — a function from the plane to the plane.
The matrix $\begin{bmatrix}2&0\\0&3\end{bmatrix}$ (input $\to$ output) sends $\langle 4, 5\rangle$ to $\langle 8, ?\rangle$. What is the output $y$?
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Row 2 is $\langle0, 3\rangle$, so the output $y = 3\cdot5 = 15$; the whole output is $\langle8, 15\rangle$.
A function from the plane to the plane
- The inputs are all vectors in the plane; the outputs are vectors in the plane too.
- So a $2\times2$ matrix is a function that reshapes the whole plane at once.
- Chaining two of them — output of one becomes input of the next — is a composition 复合.
- The composition's matrix is simply the product of the two matrices.
Chaining two matrix functions (one after another) is a composition, and its matrix is the product.
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Just like composing ordinary functions, chaining matrix functions multiplies their matrices.
Reversing the function
- If a function can be undone, its undo is the inverse function 反函数.
- For a matrix function, the inverse function is exactly the inverse matrix.
- It exists only when the determinant is non-zero — the same condition as before.
- Applying a matrix then its inverse returns every vector to where it started.
A matrix as a function
A matrix takes an input vector and returns an output vector - a function on vectors.
The inverse function of a matrix function is given by…
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The inverse matrix undoes the function — it exists only when the determinant is non-zero.
Select all true statements about matrices as functions.
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The output is a vector, not a number. The other three are correct.
Why the function view helps
- It lets you reuse function ideas: domain, range, composition, inverse.
- Composition-as-multiplication explains why transformation order matters.
- The determinant test becomes "does this function have an inverse?"
- This is the bridge from precalculus functions to linear algebra.
Not every matrix function can be reversed. If the determinant is zero, the matrix squashes the plane onto a line — many inputs share one output, so there is no inverse function. Reversibility requires a non-zero determinant.
Let $f(\vec{x}) = \begin{bmatrix}2&0\\0&3\end{bmatrix}\vec{x}$.
- Input $\langle1,1\rangle$ → output $\langle2,3\rangle$.
- Input $\langle4,5\rangle$ → output $\langle8,15\rangle$.
- Determinant $= 2\cdot3 - 0 = 6 \ne 0$, so $f$ has an inverse function that shrinks by $\tfrac12$ and $\tfrac13$.
A matrix is a function whose input and output are both vectors — a map from the plane to the plane. Chaining matrix functions is composition, and its matrix is the product. The inverse function is the inverse matrix, which exists only when the determinant is non-zero.
Write from a first output of 12
- The matrix with rows (4, 0) and (0, 2) acts on (3, 1).
- Find the first output entry.
The matrix with rows (4, 0) and (0, 2) is applied to the input (3, 1). What is the first entry of the output?
4 times 3 is 12.
What is the first output entry?
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The first diagonal entry scales the first input. Other options — 15 (the lesson output); 2 (the second output entry).