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A tensor is a representation of a multilinear function. The most familiar examples of tensors are vectors and matrices, which can be viewed as linear functions of one and two arguments, respectively.

Because a linear function is completely specified by the values it takes on a basis, it is customary to think of a tensor as a multidimensional array of these values, known as the tensor components. From this point of view, a tensor is the same as a multidimensional array, assuming one specifies the basis of the vector space on which the tensor acts.

Background: Vector Spaces

A vector space is a set of objects that can be added, subtracted, and multiplied by scalars. More formally, a vector space is a set $V$ of “vectors” and an associated set $S$ of “scalars. The scalars are almost always chosen to be the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$.

The set $V$ is equipped with an operation “$+$” such that it forms an Abelian group. Multiplication of an element of $V$ by a scalar results in another element of the space.

Example 1: real $N$-dimensional vectors

The most familiar example of a vector space is $\mathbb{R}^N$. Elements of $\mathbb{R}^N$ are ordered collections of $N$ real numbers which can be added or subtracted element-wise and multiplied by real scalars. These elements are the familiar real vectors, for example

\begin{align*} \mathbf{v} & = (0.1,\ 7.2,\ 3.0) \\ e^1 & = (1,0,0) \\ e^2 & = (0,1,0) \\ e^3 & = (0,0,1) \\ \mathbf{v} & = 0.1 e^1 + 7.2 e^2 + 3.0 e^3 \end{align*}

Example 2: Hermitian matrices

A less familiar example of a vector space is the set of $N \times N$ Hermitian matrices. The sum of difference of Hermitian matrices is also Hermitian, as is the product of a Hermitian matrix and a real scalar. Note that allowing the scalars to be complex would not result in a vector space, since the product of a Hermitian matrix with a complex number is not generally Hermitian.

Background: Dual Vectors

Given a vector space $V$, one can consider the set of linear functions on $V$. This set is denoted $V^*$. By a linear function, what is meant is a function $f \in V^*$ such that

$$ f(\alpha \mathbf{v} + \beta \mathbf{w}) = \alpha f(\mathbf{v}) + \beta f(\mathbf{w}) $$ for any $\mathbf{v}, \mathbf{w} \in V$ and scalars $\alpha, \beta$. It turns out that the set $V^$ is a vector space, and is known as the *dual space*, or the dual of the space $V$. Its elements are known as *dual vectors* or *covectors*. The reason for the term *dual* is that just as elements of $V^$ are linear functions on $V$, likewise elements of $V$ can be viewed as linear functions on $V^$. In quantum physics, Dirac bra-ket notation is used to distinguish between vectors and covectors, with "kets" being vectors and "bras" being covectors. ## Tensors Given some vector space $V$, a tensor $T$ is a multilinear function whose arguments are $r$ elements of $V^$ and $s$ elements of $V$. Such a tensor is said to be of type $(r,s)$. For example, a tensor of type $(1,1)$ is a function $T$ such that $$ T(\alpha f + \beta g, \gamma v + \delta w) = \alpha \gamma T(f,v) + \beta \gamma T(g,v) + \alpha \delta T(f,w) + \beta \delta T(g,w) $$ Say that the vector spaces $V$ (and therefore $V^$) are two-dimensional. Take ${e^1, e^2}$ to be a basis of $V$ and ${e_1, e_2}$ a basis of $V^$. Then the numbers

\begin{align} T_{11} = T(e_1, e^1) \ ;\ T_{12} = T(e_1, e^2) \\\\ T_{21} = T(e_2, e^1) \ ;\ T_{22} = T(e_2, e^2) \end{align}

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