Chapter 28. Dual Spaces

The dual space of a vector space is the vector space of all linear maps from that space to its field of scalars. If $V$ is a vector space over $F$, then its dual space is written

and is defined by

The elements of $V^*$ are called linear functionals, covectors, or linear forms. A linear functional takes a vector as input and returns a scalar. The dual space itself is a vector space under pointwise addition and scalar multiplication.

28.1 Linear Functionals

A linear functional on $V$ is a linear map

It satisfies

and

The output is a scalar, not a vector.

For example, define

by

Then $\varphi$ is linear. It takes a vector in $\mathbb{R}^3$ and returns one real number.

28.2 The Dual Space

The dual space $V^*$ is the set of all linear functionals on $V$:

If $\varphi,\psi\in V^*$, define

If $c\in F$, define

With these operations, $V^*$ is a vector space.

The zero vector of $V^*$ is the zero functional:

for every $v\in V$.

28.3 Functionals on $\mathbb{R}^n$

Every linear functional on $\mathbb{R}^n$ has the form

Equivalently,

where

Thus a linear functional on $\mathbb{R}^n$ may be represented by a row vector:

This representation depends on the standard basis.

28.4 Covectors

Vectors in $V$ and vectors in $V^*$ have different roles.

A vector $v\in V$ is an input object. A covector $\varphi\in V^*$ is a scalar-valued linear measurement.

The pairing is written

Some texts also write

The pairing takes one covector and one vector and returns a scalar. It is linear in each argument.

28.5 The Dual Basis

Let

be a basis of $V$. The dual basis is the basis

of $V^*$ defined by

Here $\delta^i_j$ is the Kronecker delta:

Thus $v^i$ extracts the $i$-th coordinate relative to the basis $B$.

28.6 Meaning of the Dual Basis

If

then

So the dual basis functional $v^i$ reads off the $i$-th coordinate of $v$.

For example, in $\mathbb{R}^3$ with the standard basis,

The dual basis consists of the coordinate projection maps.

28.7 Basis of the Dual Space

If $V$ has basis

then the dual basis

is a basis of $V^*$. Therefore

when $V$ is finite-dimensional.

Every functional $\varphi\in V^*$ can be written uniquely as

The coefficients are

Indeed, for

we have

28.8 Example in $\mathbb{R}^2$

Let $V=\mathbb{R}^2$ with standard basis

The dual basis is

Let

Then

The coordinate vector of $\varphi$ in the dual basis is

28.9 Example with a Nonstandard Basis

Let

where

We seek the dual basis

Write

The conditions are

Thus

Solving gives

So

Similarly, write

The conditions are

Solving gives

Thus

28.10 Dual Basis and Inverse Matrices

Let $B=(v_1,\ldots,v_n)$ be a basis of $\mathbb{R}^n$, and let

The dual basis functionals are the rows of

Indeed, the condition

says that the $i$-th dual row applied to the $j$-th basis column gives the $(i,j)$-entry of the identity matrix.

Thus

This is often the fastest way to compute a dual basis in coordinates.

28.11 The Natural Pairing

There is a natural pairing

defined by

This pairing is bilinear:

and scalar multiplication can be moved through either side.

The pairing connects a vector space with its dual without requiring an inner product.

28.12 Dual Space Versus Inner Product Identification

In $\mathbb{R}^n$, every vector $a$ defines a functional

This uses the standard dot product.

It is tempting to identify $V$ and $V^*$. In finite-dimensional inner product spaces, this can be done. But the identification depends on the inner product.

Without an inner product, $V$ and $V^*$ are separate spaces. A vector is not automatically a covector.

This distinction becomes important in geometry, tensor algebra, differential forms, and functional analysis.

28.13 The Double Dual

The dual of the dual space is

There is a natural map

defined by

Thus $J(v)$ is a functional on $V^*$: it takes a covector $\varphi$ and evaluates it at $v$.

If $V$ is finite-dimensional, then $J$ is an isomorphism:

This identification is canonical. It does not require choosing a basis.

28.14 Transpose of a Linear Map

Let

be a linear map. The dual map, also called the transpose or pullback, is

defined by

That is, if $\psi$ is a functional on $W$, then $T^*(\psi)$ is a functional on $V$.

For $v\in V$,

The direction reverses: $T$ goes from $V$ to $W$, while $T^*$ goes from $W^*$ to $V^*$.

28.15 Matrix of the Dual Map

Suppose $T:V\to W$ is represented by a matrix $A$ with respect to bases of $V$ and $W$. Then $T^*:W^*\to V^*$ is represented by the transpose matrix

with respect to the corresponding dual bases.

This explains the name transpose.

If

describes the original map on column vectors, then the dual map sends row functionals backward:

In column-coordinate notation for dual bases, this becomes multiplication by $A^T$.

28.16 Annihilators

Let $S\subseteq V$. The annihilator of $S$ is

It is a subspace of $V^*$.

If $U\subseteq V$ is a subspace, then $U^0$ consists of all linear functionals that vanish on $U$.

For example, in $\mathbb{R}^2$, if

then a functional

vanishes on $U$ exactly when

Thus

28.17 Dimension of an Annihilator

If $V$ is finite-dimensional and $U\subseteq V$, then

This is analogous to the dimension of an orthogonal complement.

Indeed, if

and

then a basis of $U$ can be extended to a basis of $V$. The dual basis functionals that vanish on $U$ correspond exactly to the basis vectors added outside $U$.

Thus the annihilator measures the number of independent linear conditions that ignore $U$.

28.18 Quotients and Dual Spaces

There is a natural relation between quotient spaces and annihilators:

A functional on $V/U$ corresponds to a functional on $V$ that vanishes on $U$.

Indeed, if

is linear, then

defines a linear functional on $V$, and $\varphi$ vanishes on $U$.

Conversely, if $\varphi\in U^0$, then

is well-defined.

28.19 Common Notation

28.20 Summary

The dual space $V^*$ is the vector space of all linear functionals on $V$. Its elements measure vectors by producing scalars.

The key ideas are:

Dual spaces turn vectors into objects that can be measured linearly. They are the natural home of coordinates, linear equations, transpose maps, annihilators, and many constructions that appear later in geometry and analysis.