of as numerical, but rather are indeterminates. This notation captures the expressiveness of indices and the basis-independence of index-free notation. Component-free notation A component-free treatment of tensors uses notation that emphasises that tensors do not rely on any basis, and is defined in terms of the tensor product of vector spaces. Operations There are several operations on tensors that again produce a tensor. The linear nature of tensor implies that two tensors of the same type may be added together, and that tensors may be multiplied by a scalar with results analogous to the scaling of a vector. On components, these operations are simply performed component-wise. These operations do not change the type of the tensor; but there are also operations that produce a tensor of different type. Tensor product The tensor product takes two tensors, S and T, and produces a new tensor, , whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e., which again produces a map that is linear in all its arguments. On components, the effect is to multiply the components of the two input tensors pairwise, i.e., If is of type and is of type , then the tensor product has type . Contraction Tensor contraction is an operation that reduces a type tensor to a type tensor, of which the trace is a special case. It thereby reduces the total order of a tensor by two. The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component. Components for which those two indices are different are discarded. For example, a -tensor can be contracted to a scalar through . Where the summation is again implied. When the -tensor is interpreted as a linear map, this operation is known as the trace. The contraction is often used in conjunction with the tensor product to contract an index from each tensor. The contraction can also be understood using the definition of a tensor as an element of a tensor product of copies of the space V with the space V∗ by first decomposing the tensor into a linear combination of simple tensors, and then applying a factor from V∗ to a factor from V. For example, a tensor can be written as a linear