The study of Algebraic Operations on Matrices, Vectors and Tensors is called Linear Algebra.
Tensors: Tensors are Ordered or Organised Collection of Items (which are generally Numerical Values). Each item in the collection is known as an Element. The elements are organised in Zero or More Dimensions. Tensors are used to represent Physical Entities/Quantities or Transformations to Physical Entities/Quantities.
Dimension: When we have more than one Element, they need to be ordered or organized in a certain manner to determine their Position / Location / Priority etc.
A Dimension is a Set of Indexed Locations that hold Elements of a Tensor.
Adding the First Dimension to a Tensor involves Adding at least 2 or more Indexed Locations.
Adding the Second Dimension to a Tensor involves Adding an Integer Multiple (> than 0) of as many Index Locations as there are present in the First Dimension.
Adding the Third Dimension to a Tensor involves Adding an Integer Multiple (> than 0) of as many Index Locations as there are present in the Product of Number of Indices in First Dimension and Second dimension.
Adding the Fourth Dimension to a Tensor involves Adding an Integer Multiple (> than 0) of as many Index Locations as there are present in the Product of Number of Indices in First Dimension, Second Dimension and Third Dimension.
Adding the \(N^{th}\) Dimension to a Tensor involves Adding a Integer Multiple (> than 0) of as many Index Locations as there are present in the Product of Number of Indices in \(N-1, N-2, N-3, ...\) and First Dimensions.
The Number of Locations/Elements \(K\) in a Tensor can be calculated as follows
\(K =\) (No. of Indices Per Dimension)No. of Dimensions (If all Dimensions are have same Number of Indices)
\(K =N_1 \times N_2 \times N_3 \times ... \times N_n\) (If all Dimensions are have Different Number of Indices)
where \(N_1, N_2, N_3, ..., N_n\) are the Number of Indices present in Dimensions 1,2,3,...,n respectively.
The Number of Dimensions present in a Tensor is called Rank of the Tensor.
Scalar: A Tensor with a Single Element is known as a Scalar. A Scalar does not need any ordering or organisation. Hence Scalars are also known as a Tensors with Rank 0 (Zero Dimension).
Vector: A Tensor having 2 or More Elements organised in a Single Row or a Single Column is known as a Vector. Hence Vectors are also known as a Tensors with Rank 1 (1 Dimension)
Matrix: A Tensor having Elements Organised Across a Table of Rows and Columns is known as a Matrix. Matrices having only a Single Row or a Single Column are same as Vectors. However when refering to Matrices, both Dimensions are mentioned. Matrices are also known as a Tensors with Rank 2 (2 Dimensions)
Cube/Cuboid: A Tensor having Elements Organised Accross More than 1 Table of Rows and Columns is known as a Cube/Cuboid. Hence Cubes/Cuboids are also known as a Tensors with Rank 3 (3 Dimensions)