What is a dual space?

Question

I've started studying differential geometry by myself and I ran into dual spaces in a section on 1-forms. I'm not very well versed in linear algebra so any help is much appreciated.

Answer 1

I recommend a excellent reading about differential forms with some previous results in linear algebra, in my point-view.

1: Differential Forms, by Henri Cartan. This book is ideal for understand differential forms in various contexts, for example, Cartan develops the theory of forms in space of finite and infinite dimension.

And for differential forms in geometry, you can consult:

2: Differential Forms, by Manfredo do Carmo. This book is ideal to learn the concepts of differential forms to apply in Differential Geometry.

So a quick reading of Manfredo's book will be great for you and your doubts, but I recommend, principally for geometry study, a reading in Cartan's book.

Answer 2

Roughly, a 1-form $f: V \rightarrow K$, is a function that eats a vector and spits out a scalar, and is also linear:

$$f(\alpha \textbf{v} + \beta \textbf{w}) = \alpha f(\textbf{v}) + \beta f(\textbf{w})$$

For a quick introduction, you can check out the wikipedia page. I particularly like the visualization they have of envisioning 1-forms in finite dimensions as stacks of oriented $(n-1)$-dimensional hyperplanes. When a vector is paired with a 1-form, the resulting scalar is simply the number of planes that the vector "pierces". Larger 1-forms are represented by denser stacks, and produce larger scalars for the same vector.

You can show that the collection of all 1-forms over a vector space $V$ forms another vector space (we can meaningfully add 1-forms, multiply them by scalars, write them as linear combinations of other 1-forms in a basis [proof], etc.). This new vector space of 1-forms is called the dual space, $V^*$.

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One particularly interesting fact about 1-forms is that they obey covariant transformation laws under a change of basis, whereas traditional vectors obey contravariant transformation rules. [wikipedia link]

Vectors are contravariant because their components transform in the opposite way that the basis does. For example, shrinking the basis vectors makes a vector's components grow larger; also, rotating the basis clockwise makes the components of a vector rotate counter-clockwise.

1-forms are covariant because their components transform in the same ways as the basis. Shrinking the basis vectors makes a 1-form's components shrink as well (the stacks of planes appear farther apart, and the 1-form is therefore smaller).

Answer 3

In the context of vector spaces, the dual space is a space of linear "measurements". When a dual vector $f$ acts on a vector $v$, the scalar output $f(v)$ provides information about $v$; in particular, it gives you something about a coordinate of $v$ in some direction. A vector $v$ can be reconstructed from knowing all the values $f(v)$ for $f$ in the dual space.

As a rough example from engineering, consider vector space $V$ the space of continuous, periodic functions on $[0, 2\pi]$. Then there are certain dual vectors $f_k$ which returns how much of the function $v(x) \in V$ contains a frequency $k$. So if $v(x)$ was a sound wave, then $f_k(v)$ tells you whether the musical note of frequency $k$ is in the sound. Then the entire sound wave $v(x)$ can be reconstructed from knowing $f_k(v)$ for all $k$, since that would tell you all the notes in the sound.

And of course, $f_k(v)$ is going to be the $k$th Fourier coefficient of $v$.