Matrix of a Transformation
Columns are images of basis vectors.
Key idea
Columns are images of basis vectors.
In depth
The columns of the matrix are the images of the standard basis vectors: A = [T(e₁) T(e₂) ...]. Once you know where the basis goes, you know where every vector goes.
Worked example
If T sends e₁ to (2, 1) and e₂ to (0, 3), A = [2 0; 1 3].
Real-world uses
- Representing a robot arm movement or a camera rotation as a matrix
Common mistake
Putting images in rows instead of columns.
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 Matrix of a Transformation?
Columns are images of basis vectors. It is a topic at the advanced level in Linear Algebra.
Where is Matrix of a Transformation used in real life?
Real-world examples: Representing a robot arm movement or a camera rotation as a matrix.
What should I learn before Matrix of a Transformation?
Make sure you are comfortable with Linear Transformations, the previous topic in Linear Algebra.
What is a common mistake with Matrix of a Transformation?
Putting images in rows instead of columns.
How can I practise Matrix of a Transformation?
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 Matrix of a Transformation?
Move on to Rotation and Reflection Matrices 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.