Scaling & Stretching — Scaling and Stretching Along Axes - Linear Algebra

SECTION 1

Historical Context & Motivation

People have been stretching and scaling shapes for thousands of years. Ancient map-makers needed to shrink enormous landscapes onto small scrolls. Renaissance artists used grids to enlarge tiny sketches into massive murals. In each case, the key idea was the same: multiply lengths in one or more directions to change the size of a figure without losing its basic structure.

As mathematics grew more formal, scholars began to express these ideas using numbers and symbols. Scaling — the process of multiplying coordinates by a constant — became one of the simplest yet most powerful operations in what we now call linear algebra. Today, every time you pinch-to-zoom on a phone, resize an image, or watch a 3-D movie, scaling transformations are working behind the scenes.

~300 BC • Euclid's Elements

Euclid described similar figures — shapes with the same proportions but different sizes — laying the geometric groundwork for scaling.

1637 • Descartes' Coordinate System

René Descartes introduced the x-y coordinate plane, making it possible to describe scaling with numbers and equations.

1850s • Cayley & Matrices

Arthur Cayley formalized matrix algebra, giving mathematicians a compact way to write scaling and other transformations.

1990s–Today • Computer Graphics Revolution

Video games, movies, and phone apps rely on scaling matrices millions of times per second to render images on screen.

The central question this lesson explores is: What happens to a point, a shape, or an entire coordinate plane when we multiply the x-coordinates, the y-coordinates, or both by some number? Understanding this will unlock a whole family of transformations you can describe with simple matrices.

SECTION 2

Core Principles & Definitions

Before we dive into calculations, let's nail down the vocabulary. A linear transformation is a rule that moves every point in a plane (or in space) to a new location while keeping straight lines straight and the origin fixed. Scaling along the axes is one of the most straightforward types of linear transformation: you simply multiply coordinates by constants called scale factors.

Scale Factor

A number you multiply a coordinate by. If the scale factor is 3, every x- or y-value is tripled. If it is 0.5, every value is halved.

Horizontal Scaling

Multiplying only the x-coordinate by a scale factor. This stretches or compresses a shape left-to-right without changing its height.

Vertical Scaling

Multiplying only the y-coordinate by a scale factor. This stretches or compresses a shape up-and-down without changing its width.

Uniform Scaling

Using the same scale factor for both x and y. The shape grows or shrinks proportionally, keeping its original proportions.

Non-Uniform Scaling

Using different scale factors for x and y. The shape stretches more in one direction, which can turn circles into ellipses.

✦

KEY TAKEAWAY

Think of scaling like stretching a rubber sheet that is pinned at the origin. If you pull it to the right, only the horizontal distances grow — that's horizontal scaling. If you pull it upward, only the vertical distances grow — that's vertical scaling. Pull it equally in both directions and everything gets bigger without changing shape — that's uniform scaling.

SECTION 3

Visual Explanation

The diagram below shows a unit square (a 1 × 1 square whose corners sit at the origin) before and after three different scaling transformations. Notice how the shape changes depending on which axis is scaled and by how much.

Left: the original 1 × 1 unit square. Center: horizontal scaling by a factor of 2 doubles the width but leaves the height unchanged. Right: vertical scaling by a factor of 3 triples the height but leaves the width unchanged.

In the center panel, every x-coordinate has been doubled while the y-coordinates stay the same. The point (1, 1) moved to (2, 1). In the right panel, every y-coordinate has been tripled while x-coordinates stay put, so (1, 1) moved to (1, 3). The dashed outline of the original square makes it easy to compare the before and after. Notice that the origin always stays fixed — it's the anchor point of every scaling transformation.

SECTION 4

Mathematical Framework

We can describe axis-aligned scaling with a neat formula. If we have a point (x, y) and we want to scale the x-coordinate by a factor a and the y-coordinate by a factor b, the new point lands at (a × x, b × y). In linear algebra, we write this as a matrix multiplication.

SCALING TRANSFORMATION

[ x' ] [ a 0 ] [ x ] [ a × x ] [ y' ] = [ 0 b ] [ y ] = [ b × y ]

Here, a is the horizontal scale factor, b is the vertical scale factor, (x, y) is the original point, and (x', y') is the new point after the transformation.

The 2 × 2 matrix with a in the top-left corner and b in the bottom-right corner is called a diagonal matrix. The zeros in the off-diagonal spots mean that x and y don't mix — each axis is scaled independently.

HORIZONTAL STRETCH ONLY

[ x' ] [ a 0 ] [ x ] [ a × x ] [ y' ] = [ 0 1 ] [ y ] = [ y ]

When b = 1, the y-coordinate doesn't change — only the x-direction is scaled.

VERTICAL STRETCH ONLY

[ x' ] [ 1 0 ] [ x ] [ x ] [ y' ] = [ 0 b ] [ y ] = [ b × y ]

When a = 1, the x-coordinate doesn't change — only the y-direction is scaled.

UNIFORM SCALING

[ x' ] [ k 0 ] [ x ] [ k × x ] [ y' ] = [ 0 k ] [ y ] = [ k × y ]

When a = b = k, both axes are scaled by the same factor. The shape grows or shrinks proportionally.

💡

What about negative scale factors?

If a scale factor is negative, the transformation flips (reflects) the shape across the opposite axis in addition to stretching it. For example, a = −1 and b = 1 reflects every point across the y-axis. We'll focus on positive scale factors in this lesson, but keep this in mind for later!

SECTION 5

Detailed Breakdown — How Scale Factors Change a Shape

The value of a scale factor tells you exactly what happens to a shape along that axis. Scale factors greater than 1 stretch the shape, factors between 0 and 1 compress it, and a factor of exactly 1 leaves it unchanged. The diagram below shows all three cases applied to a circle on the x-axis.

Left: a scale factor of 0.5 compresses the circle into a tall, narrow ellipse. Center: a scale factor of 1 leaves the circle unchanged. Right: a scale factor of 1.75 stretches the circle into a wide ellipse.

Scale Factor Range	Effect Along That Axis	Example
k > 1	Stretch — the shape gets longer in that direction.	k = 3 triples every distance from the origin.
k = 1	No change — the axis is unaffected.	The identity: nothing moves.
0 < k < 1	Compress — the shape gets shorter in that direction.	k = 0.25 shrinks distances to one-quarter.
k = 0	Collapse — the shape flattens onto the other axis.	a = 0 squashes all x-values to 0.

How different scale factors affect a shape along one axis

An important detail: the area of any shape after scaling equals the original area multiplied by |a × b|. So if you double the width (a = 2) and triple the height (b = 3), the new area is 2 × 3 = 6 times the original area. This product |a × b| is actually the determinant of the scaling matrix — a concept you'll explore more deeply in later courses.

SECTION 6

Worked Example

Let's work through a complete example. A triangle has vertices at A(1, 2), B(3, 1), and C(2, 4). We want to apply a scaling transformation that stretches the x-axis by a factor of 2 and compresses the y-axis by a factor of 0.5.

Scaling a Triangle with a = 2 and b = 0.5

1
Step 1 — Write the Scaling Matrix

Place the horizontal scale factor a = 2 in the top-left and the vertical scale factor b = 0.5 in the bottom-right, with zeros elsewhere: S = [ 2 0 ; 0 0.5 ].

S = [ 2 0 ; 0 0.5 ]
2
Step 2 — Transform Vertex A(1, 2)

Multiply the matrix by the column vector for A: x' = 2 × 1 = 2, and y' = 0.5 × 2 = 1.

A' = (2, 1)
3
Step 3 — Transform Vertex B(3, 1)

Apply the same matrix: x' = 2 × 3 = 6, and y' = 0.5 × 1 = 0.5.

B' = (6, 0.5)
4
Step 4 — Transform Vertex C(2, 4)

Again: x' = 2 × 2 = 4, and y' = 0.5 × 4 = 2.

C' = (4, 2)
5
Step 5 — Check the Area Change

The determinant of the scaling matrix is a × b = 2 × 0.5 = 1. This means the area of the triangle hasn't changed — the horizontal stretch was exactly canceled out by the vertical compression! The shape is wider and flatter, but its area is the same.

Area factor = |a × b| = |2 × 0.5| = 1 (area unchanged)

🔍

NOTICE THE PATTERN

Each vertex is transformed independently. You never need to worry about angles or side lengths during the multiplication — just plug each point's coordinates into the formula and let the matrix do the work.

SECTION 7

Scaling vs. Other Transformations

Scaling is only one type of linear transformation. It's helpful to see how it stacks up against the other transformations you might encounter. The table below compares scaling with rotations, reflections, and shears.

Transformation	What It Does	Preserves Shape?	Preserves Area?
Uniform Scaling	Enlarges or shrinks equally in all directions	Yes	No (unless k = ±1)
Non-Uniform Scaling	Stretches differently along x and y	No	No (unless \|a × b\| = 1)
Rotation	Spins the shape around the origin	Yes	Yes
Reflection	Flips the shape across a line	Yes	Yes
Shear	Tilts the shape like a leaning stack of cards	No	Yes

Comparison of common 2-D linear transformations

⭐

WHY SCALING IS SPECIAL

Scaling is the simplest transformation to compute because the matrix is diagonal — x and y never mix. Rotations and shears have non-zero off-diagonal entries, which means the x-output depends on y and vice versa. This independence makes scaling matrices easy to invert, combine, and analyze.

SECTION 8

Connection to Advanced Theory

Axis-aligned scaling is actually the starting point for some powerful ideas in higher-level linear algebra. When you take a more advanced course, you'll learn that any linear transformation — rotation, shear, or some complicated combination — can be broken down into a rotation, followed by axis-aligned scaling, followed by another rotation. This powerful idea is called the Singular Value Decomposition (SVD), and it shows that scaling is at the heart of every linear transformation.

This Lesson	What Comes Next
Scale factors a and b on the diagonal	Eigenvalues — the natural scale factors of any square matrix
The x- and y-axes are the special directions	Eigenvectors — the special directions that only get scaled (not rotated) by a matrix
Determinant = a × b (area scale factor)	Determinant of any matrix — tells you how much the transformation changes area (or volume)
Diagonal 2 × 2 matrix	Diagonalization — rewriting a complex matrix as a scaling matrix in a new coordinate system

How scaling concepts extend to advanced linear algebra

In short, mastering axis-aligned scaling gives you a concrete, visual foundation for the abstract concepts that power machine learning, computer graphics, data science, and engineering. Every one of those fields uses eigenvalues, eigenvectors, and the SVD every day — and they all trace back to the simple idea of multiplying a coordinate by a scale factor.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A scaling matrix has a = 1 and b = 1. What does this transformation do to any point in the plane? Explain why.

Show answer

This transformation does nothing — every point stays exactly where it is. The matrix [ 1 0 ; 0 1 ] is called the identity matrix. When you multiply any point (x, y) by it, you get (1 × x, 1 × y) = (x, y). A scale factor of 1 means 'keep the same size,' so applying it to both axes leaves the entire plane unchanged.

Question 2: BASIC CALCULATION

Apply the scaling matrix S = [ 3 0 ; 0 2 ] to the point P(4, −1). What are the coordinates of the image point P'?

Show answer

Multiply each coordinate by its scale factor: x' = 3 × 4 = 12 and y' = 2 × (−1) = −2. So the image point is P'(12, −2). The x-coordinate tripled and the y-coordinate doubled. Notice that the point moved farther from the origin in both directions because both scale factors are greater than 1.

Question 3: INTERMEDIATE

A rectangle has corners at (0, 0), (4, 0), (4, 3), and (0, 3). After a scaling transformation, the new corners are at (0, 0), (2, 0), (2, 9), and (0, 9). Find the scale factors a and b, and determine the ratio of the new area to the original area.

Show answer

Compare corresponding coordinates. The x-coordinate 4 became 2, so a = 2 ÷ 4 = 0.5. The y-coordinate 3 became 9, so b = 9 ÷ 3 = 3. The original area is 4 × 3 = 12 square units. The new area is 2 × 9 = 18 square units. The ratio of new area to old area is 18 ÷ 12 = 1.5, which equals |a × b| = |0.5 × 3| = 1.5. This confirms that the determinant of the scaling matrix gives you the area scale factor.

Question 4: APPLIED

A game designer has a character sprite that is 64 pixels wide and 128 pixels tall. She wants to display it at 1.5× width and 0.75× height for a special animation frame. Write the scaling matrix she should use, and find the new pixel dimensions of the sprite.

Show answer

The scaling matrix is S = [ 1.5 0 ; 0 0.75 ]. The new width is 1.5 × 64 = 96 pixels and the new height is 0.75 × 128 = 96 pixels. Interestingly, the originally non-square sprite becomes a perfect square after this transformation. The area scale factor is |1.5 × 0.75| = 1.125, so the new sprite covers 12.5% more total pixels than the original.

Question 5: CRITICAL THINKING

If you apply two scaling transformations one after the other — first S₁ = [ a₁ 0 ; 0 b₁ ] and then S₂ = [ a₂ 0 ; 0 b₂ ] — what single scaling matrix would produce the same result? Does the order in which you apply them matter? Justify your answer.

Show answer

Applying S₁ first gives (a₁ × x, b₁ × y). Then applying S₂ gives (a₂ × a₁ × x, b₂ × b₁ × y). The combined matrix is S = [ a₁ × a₂ 0 ; 0 b₁ × b₂ ]. If you reverse the order and apply S₂ first and then S₁, you get (a₁ × a₂ × x, b₁ × b₂ × y), which is exactly the same result because multiplication of numbers is commutative. Therefore, the order does not matter for axis-aligned scaling — unlike rotations or shears, which generally do not commute. This is a special property of diagonal matrices: they always commute with each other.

SUMMARY

Lesson Summary

Summary

Scaling along the axes is a linear transformation that multiplies the x-coordinate by a horizontal scale factor a and the y-coordinate by a vertical scale factor b. The transformation is written as the diagonal matrix [ a 0 ; 0 b ]. Scale factors greater than 1 stretch a shape along that axis, factors between 0 and 1 compress it, and a factor of 1 leaves it unchanged. The origin always stays fixed.

The product |a × b| gives the area scale factor (determinant) of the transformation. When a = b, the scaling is uniform and preserves the shape's proportions. When a ≠ b, the scaling is non-uniform and distorts the shape. This simple concept connects directly to eigenvalues, eigenvectors, and the Singular Value Decomposition in advanced linear algebra.