Linear Maps Between Spaces
Kernel, image and isomorphism.
Key idea
Kernel, image and isomorphism.
In depth
A linear map T: V → W can be between different spaces. Its matrix depends on chosen bases for V and W. Kernel and image generalise null space and column space.
Worked example
Differentiation maps polynomials of degree ≤ 3 to degree ≤ 2 and is linear, with kernel the constants.
Real-world uses
- Describing how data is compressed or reconstructed in coding and signal processing
Common mistake
Forgetting that the matrix depends on the basis.
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 Maps Between Spaces?
Kernel, image and isomorphism. It is a topic at the expert level in Linear Algebra.
Where is Linear Maps Between Spaces used in real life?
Real-world examples: Describing how data is compressed or reconstructed in coding and signal processing.
What should I learn before Linear Maps Between Spaces?
Make sure you are comfortable with Gershgorin Circle Theorem, the previous topic in Linear Algebra.
What is a common mistake with Linear Maps Between Spaces?
Forgetting that the matrix depends on the basis.
How can I practise Linear Maps Between Spaces?
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 Maps Between Spaces?
Move on to Dual Space 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.