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Why is the notation for the dual space $V^*:V\rightarrow F$ often given as an introduction? Wouldn't it be simpler to just say vectors are column vectors and covectors, elements of the dual space, are row vectors? Is writing this definition only for rigorous purposes, or is there something greater given by this introduction? I don't really understand the dual space myself.

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    $\begingroup$ The notion of ‘dual space’ is not retrained to finite dimensional spaces and vectors are not matrices. Further, how you interpret the bidual $\;V^{**} : V^*\to F$ in terms of vectors or covectors? $\endgroup$
    – Bernard
    Commented Apr 27, 2018 at 0:33

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Well, vectors aren't "column vectors". The row-vector vs column-vector intuition is good to fall back on, but we work at a level beyond this. A vector is nothing more than an element of any vector space. No column needed. If this is what a vector is, then what should be the analog of a row vector? Well, you can dot-product a row vector and column vector to get an element of the base field. That is, each row vector acts as a linear map from column vectors to the base field. Moreover, all such maps are of this form. It then seems that we should say that a dual vector is just a linear map from the vector space in question to the base field.

In this way, the linear map definition is more important, since it gives us row vectors in the extra-special case when we have column vectors, but it works for any vector space (eg: the vector space of real polynomials).

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That notation isn't really correct.

If $V$ is a vector space of a field $\mathbb{F}$, then $V^* = \{\phi: V\rightarrow \mathbb{F}| \ \phi\text{ is linear} \}$. In other words, it's the set of linear functions that take in vectors from $V$ and return scalars in $\mathbb{F}$. For example, if $V = \mathbb{R}^3$ and $\mathbb{F} = \mathbb{R}$, then $\phi_w: v \mapsto v\cdot w $ for some particular $w\in\mathbb{R}^3$ is a member of $\left(\mathbb{R}^3\right)^*$, since it takes in a vector $v\in\mathbb{R}^3$ and computes the scalar product with $w$, returning a scalar $v\cdot w \in \mathbb{R}$.

The reason we define the dual space in terms of functions is because the dual space is a set of functions. Any element in the dual space is a function. It is also a vector, because the dual space of a vector space is a vector space (this requires proof).

If you're learning about dual spaces, then you need to dispel your notion that a vector is an arrow, or that a vector is a row or column matrix, or that a vector is any one thing in particular. In fact, the word vector isn't the name of an object, it's a description. Something is a vector when it satisfies the requirements of being a vector, just like something is red when it satisfies the requirements of being red. A car can be red, a person's hair can be red, dirt can be red, lots of things can be red. To make a blanket statement about all things that are red would be ridiculous, as the only thing they have in common is that they are red. The same is true for vectors, the only thing they have in common is that they are members of a vector space.

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    $\begingroup$ Now that I get it feel this answer explains it better. Part of the problem was I thought I understood the mapping notation but didn't. $\endgroup$ Commented Dec 21, 2018 at 22:26

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