It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives -1?
Definition[ edit ] Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction.
As multidimensional arrays[ edit ] A tensor may be represented as a potentially multidimensional array although a multidimensional array is not necessarily a representation of a tensor, as discussed below with regard to holors. Just as a vector in an n- dimensional space is represented by a one-dimensional array of length n with respect to a given basisany tensor with respect to a basis is represented by a multidimensional array.
The numbers in the multidimensional array are known as the scalar components of the tensor or simply its components. They are denoted by indices giving their position in the array, as subscripts and superscriptsfollowing the symbolic name of the tensor.
For example, the components of an order 2 tensor T could be denoted Tijwhere i and j are indices running from 1 to n, or also by T j i. Whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below.
Thus while Tij and T j i can both be expressed as n by n matrices, and are numerically related via index jugglingthe difference in their transformation laws indicates it would be improper to add them together. The total number of indices required to identify each component uniquely is equal to the dimension of the array, and is called the order, degree or rank of the tensor.
However, the term "rank" generally has another meaning in the context of matrices and tensors. Just as the components of a vector change when we change the 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 vectorswhere the new basis vectors e.If given the component form to represent a hypothetical vector, how would one represent this vector as a linear combination of the standard unit vectors, (i) and (j)?
Because (i) represents a vector in the (x) direction, multiplying V₁ by (i) would give the component form. This will give us the unit vector in the same direction as a given vector u To write the linear combination, just take out the horizontal/vertical component of .
The initial and terminal points of a vector are barnweddingvt.com a linear combination of the standard unit vectors i and j. Initial Point Terminal Point. Writing a Vector as a Linear Combination of Other Vectors Sometimes you might be asked to write a vector as a linear combination of other vectors.
This requires the same work as above with one more step. Euler's identity seems baffling: It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives .
The Ohio State University linear algebra midterm exam problem and its solution is given. Express a vector as a linear combination of other three vectors.