LINEAR ALGEBRA (Part-1) - Scalars, Vectors, Matrices and Tensors

LINEAR ALGEBRA (Part-1)

Scalars, Vectors, Matrices and Tensors

Scalars

Scalars are a single number. like 5 or 8 or 9 etc.
you can say age=23 here 23 is a scalar value.

Vectors

Vectors are a series/array of elements. They are represented between a square bracket with spaces.
Like below.

ages	=	[
				23
				25
				33
				54
		]

Here,

		[
				23
				25
				33
				54
		]

is a Column Vector of ages. The Row Vector of ages will look like ages = [23 25 33 54] each element of a vector can be represented by a subscript number like ages₁=23 or ages₃=33 etc. the numbers of a vector do not have to be of same thing like.

box	=	[
				2
				4
				8
				"wood"
		]

here the box vector represents,

box1 = height = 2 inch

box2 = width = 4 inch

box3 = weight = 8 gram

box4 = made of = wood

Matrix

Matrix is a 2D array of elements.
Like below.

box	=	[
				2		3		2		4
				4		4		5		2
				8		12		500		8
				"wood"		"wood"		"iron"		"wood"
		]

here box matrix represents 4 vector/boxes.

box1	=	[
				2
				4
				8
				"wood"
		]

box2	=	[
				3
				4
				12
				"wood"
		]

box3	=	[
				2
				5
				500
				"iron"
		]

box4	=	[
				4
				2
				8
				"wood"
		]

this is a 4X4 matrix as 4 vectors each has 4 values.

Matrix Transpose

Lets represent a 3X2 matrix.
Like below.

A	=	[
				A11		A12
				A21		A22
				A31		A32
		]

so the transpose of A matrix will be

A	=	[
				A11		A21	A31
				A12		A22	A32
		]

Transpose of a 3X2 matrix will be 2X3 matrix.

Tensors

More then 2D array of elements are called Tensors.
Like below.

box =	[
		[
				2		3		2		4
				4		4		5		2
				8		12		500		8
				"wood"		"wood"		"iron"		"wood"
		],
		[
				2		3		2		4
				4		4		5		2
				8		12		500		8
				"wood"		"wood"		"iron"		"wood"
		]
	]

SOURCE

Deep Learning (Ian Goodfellow, Yoshua Bengio, Aaron Courville)