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A tensor may be represented as a (potentially multidimensional) array. Just as a [[Vector space|vector]] in an {{mvar|n}}-[[dimension (vector space)|dimensional]] space is represented by a [[multidimensional array|one-dimensional]] array with {{mvar|n}} components with respect to a given [[Basis (linear algebra)#Ordered bases and coordinates|basis]], any tensor with respect to a basis is represented by a multidimensional array. For example, a [[linear operator]] is represented in a basis as a two-dimensional square {{math|''n'' × ''n''}} array. The numbers in the multidimensional array are known as the ''components'' of the tensor. They are denoted by indices giving their position in the array, as [[subscript and superscript|subscripts and superscripts]], following the symbolic name of the tensor. For example, the components of an order {{math|2}} tensor {{mvar|T}} could be denoted {{math|''T''<sub>''ij''</sub>}} , where {{mvar|i}} and {{mvar|j}} are indices running from {{math|1}} to {{mvar|n}}, or also by {{math|''T''{{thinsp}}{{su|lh=0.8|b=''j''|p=''i''}}}}. Whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. Thus while {{math|''T''<sub>''ij''</sub>}} and {{math|''T''{{thinsp}}{{su|lh=0.8|b=''j''|p=''i''}}}} can both be expressed as ''n''-by-''n'' matrices, and are numerically related via [[Raising and lowering indices|index juggling]], the difference in their transformation laws indicates it would be improper to add them together.
The total number of indices ({{mvar|m}}) required to identify each component uniquely is equal to the ''dimension'' or the number of ''ways'' of an array, which is why
Just as the components of a vector change when we change the [[basis (linear algebra)|basis]] of the vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a ''transformation law'' that details how the components of the tensor respond to a [[change of basis]]. The components of a vector can respond in two distinct ways to a [[change of basis]] (see ''[[Covariance and contravariance of vectors]]''), where the new [[basis vectors]] <math>\mathbf{\hat{e}}_i</math> are expressed in terms of the old basis vectors <math>\mathbf{e}_j</math> as,
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