Linear Transformations
Functions preserving addition and scalar multiplication.
Key idea
Functions preserving addition and scalar multiplication.
In depth
A map T is linear if T(u + v) = T(u) + T(v) and T(cu) = cT(u). Such maps preserve lines and the origin. All of them (in finite dimensions) can be represented by matrices.
Worked example
T(x, y) = (2x, x + y) is linear. T(x, y) = (x + 1, y) is not, because it moves the origin.
Formula
Real-world uses
- Rotating, stretching or projecting images in graphics and mapping data in machine learning
Common mistake
Thinking any straight-line formula is linear. Adding a constant breaks it.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Linear Transformations?
Functions preserving addition and scalar multiplication. It is a topic at the advanced level in Linear Algebra.
What is the formula for Linear Transformations?
The key formula is T(au+bv) = aT(u)+bT(v).
Where is Linear Transformations used in real life?
Real-world examples: Rotating, stretching or projecting images in graphics and mapping data in machine learning.
What should I learn before Linear Transformations?
Make sure you are comfortable with Adjugate Matrix, the previous topic in Linear Algebra.
What is a common mistake with Linear Transformations?
Thinking any straight-line formula is linear. Adding a constant breaks it.
How can I practise Linear Transformations?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Linear Transformations?
Move on to Matrix of a Transformation next.
Related advanced topics in Linear Algebra
- Dot Product – Sum of component products.
- Angle Between Vectors – Cosine from the dot product.
- Orthogonality – Vectors are orthogonal when their dot product is 0.
- Projection – Component of one vector along another.
- Cross Product – A vector perpendicular to two vectors in 3D.
- Lines and Planes in Vector Form – Parametric and normal equations.