Transpose
Swap rows and columns.
Key idea
Swap rows and columns.
In depth
Aᵀ swaps rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. Key rules: (AB)ᵀ = BᵀAᵀ (order reverses), (Aᵀ)ᵀ = A. A matrix equal to its transpose is symmetric.
Worked example
[1 2 3; 4 5 6]ᵀ = [1 4; 2 5; 3 6].
Formula
Real-world uses
- Rearranging data tables for analysis
- Computing least squares solutions in statistics
Common mistake
Forgetting that the order reverses when transposing a product.
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 Transpose?
Swap rows and columns. It is a topic at the advanced level in Linear Algebra.
What is the formula for Transpose?
The key formula is (AB)ᵀ = BᵀAᵀ.
Where is Transpose used in real life?
Real-world examples: Rearranging data tables for analysis; Computing least squares solutions in statistics.
What should I learn before Transpose?
Make sure you are comfortable with Identity Matrix, the previous topic in Linear Algebra.
What is a common mistake with Transpose?
Forgetting that the order reverses when transposing a product.
How can I practise Transpose?
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 Transpose?
Move on to Special 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.