Tensors Introduction
Multilinear maps generalising vectors and matrices.
Key idea
Multilinear maps generalising vectors and matrices.
In depth
A tensor is a multilinear object that generalises scalars, vectors and matrices: order 0, 1, 2 and above. Components transform in specific ways under a change of coordinates. They appear in physics (stress, curvature) and machine learning (multidimensional arrays).
Worked example
A matrix is an order-2 tensor with two indices.
Real-world uses
- Describing stress in materials, curvature in general relativity and data arrays in deep learning
Common mistake
Thinking a tensor is just a multidimensional array, ignoring transformation rules.
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 Tensors Introduction?
Multilinear maps generalising vectors and matrices. It is a topic at the expert level in Linear Algebra.
Where is Tensors Introduction used in real life?
Real-world examples: Describing stress in materials, curvature in general relativity and data arrays in deep learning.
What should I learn before Tensors Introduction?
Make sure you are comfortable with Direct Sums, the previous topic in Linear Algebra.
What is a common mistake with Tensors Introduction?
Thinking a tensor is just a multidimensional array, ignoring transformation rules.
How can I practise Tensors Introduction?
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 Tensors Introduction?
Move on to Linear Algebra over Finite Fields next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.