Linear Transformation

TL;DR

• Operations that change a function’s position or size while preserving its linear relationships.
• Common forms include translation (shifting) and scaling (stretching/shrinking).
• Outputs of linear transformations can be expressed as linear combinations of the inputs.

Definition

Linear transformations are mathematical operations that preserve the linearity of a function. In other words, they maintain the relationship between the input and output of a function, such that the output of a linear transformation can be represented as a linear combination of the inputs.

Explanation

Linear transformations manipulate functions in ways that keep linear structure intact. Two common types are described in the source:

• Translation (shifting): moves the points of a function along a given vector.
• Scaling (stretching or shrinking): changes the size of a function by a given factor.

These transformations make it possible to modify functions in predictable ways—for example, to align a function with a specific coordinate system or to make it easier to compare with other functions. Linear transformations are also applied in geometry to rotate, reflect, or shear shapes.

Examples

Translation (shifting)

If we have $f(x) = x$ we can translate it by 2 units to the right by adding 2 to each x-value, resulting in $f(x) = x + 2.$ According to the source, this new function will have the same slope and y-intercept as the original, but will be shifted to the right by 2 units.

Scaling (stretching/shrinking)

If we have $f(x) = x$ we can scale it by a factor of 2 by multiplying each x-value by 2, resulting in $f(x) = 2x.$ This new function will have the same y-intercept as the original, but will have a steeper slope and a greater range of values.

Use cases

• Aligning a function with a specific coordinate system by shifting it.
• Scaling a function to make it easier to compare to other functions.
• Geometry operations such as rotating, reflecting, or shearing shapes.
• Applications in data analysis, graphing, and engineering.

Related terms

• Linearity
• Linear combination
• Translation
• Scaling
• Rotation
• Reflection
• Shear
• Slope
• Y-intercept