Dual Space
Space of linear functionals on V.
Key idea
Space of linear functionals on V.
In depth
The dual space V* consists of all linear functionals f: V → scalars. Each basis of V gives a dual basis. In an inner product space, every functional is an inner product with some vector.
Worked example
The map (x, y) ↦ 3x − y is a functional on R², represented by the row vector (3, −1).
Real-world uses
- Describing measurements and read-outs of a system as functions in physics and statistics
Common mistake
Mixing vectors and functionals; they transform differently under a change of 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 Dual Space?
Space of linear functionals on V. It is a topic at the expert level in Linear Algebra.
Where is Dual Space used in real life?
Real-world examples: Describing measurements and read-outs of a system as functions in physics and statistics.
What should I learn before Dual Space?
Make sure you are comfortable with Linear Maps Between Spaces, the previous topic in Linear Algebra.
What is a common mistake with Dual Space?
Mixing vectors and functionals; they transform differently under a change of basis.
How can I practise Dual Space?
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 Dual Space?
Move on to Quotient Spaces 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.