Scalars:
A scalar is only a single quantity. It’s the easiest type of information, having solely magnitude with none route. Scalars are sometimes utilized in equations and calculations. Examples of scalars embrace temperature, mass, and pace. We normally give scalars lowercase italic variable names.
Instance: “Let’s s ∈ ℝ”
Properties
- Magnitude
- Arithmetic Operations
Vector:
Vector is an array of numbers organized so as. Not like scaler vector has each magnitude and route. We will establish every particular person quantity by index. Sometimes we give vector daring lowercase identify. Ingredient of vector with italic typeface, in v 1st factor as v₁ 2nd factor as v₂ & so on. To point the kind of numbers within the vector {ℝ, ℕ, ℤ, and so on.} and the dimension of vector, we use notation like ℝⁿ & ℝ³ n/3-Dimensional vector containing actual numbers.
We will consider vector as level in n-Dimensional house with every factor giving coordinate alongside totally different axis.
Properties
- Magnitude and Path
- Illustration: Vectors could be represented graphically as arrows or numerically as arrays.
- Operations: Vectors could be added and subtracted, and they are often scaled by a scalar. The dot product and cross product are particular vector operations utilized in varied functions.
Matrices:
A Matrices is 2-D array of quantity recognized by two indices as an alternative of only one. They’re utilized in varied fields, together with laptop graphics, the place they assist in transformations and rotations, and in linear algebra for fixing methods of equations. We normally give matrices uppercase daring typeface. reminiscent of A, however To point the kind of numbers & peak(3) and width(3), we are saying like “ A ∈ ℕ³*³ ” & factor in italics like A₂₁ or ∱(A)₂₁.
Properties
- Rows and Columns
- Operations: Matrices could be added, subtracted, and multiplied. Matrix multiplication just isn’t commutative, which means the order of multiplication issues.
- Determinant and Inverse: properties of matrices which might be utilized in fixing linear equations and in transformations.
Tensor:
Tensors generalize the ideas of scalars, vectors, and matrices to increased dimensions. A tensor is actually an n-dimensional array of numbers. Tensors are used extensively in machine studying, particularly in deep studying frameworks like TensorFlow & PyTorch.
In layman’s phrases, tensors symbolize information in one-dimensional to n-dimensional areas, extending the ideas of vectors and matrices.
Properties
- Multi-Dimensional: Tensors can have a number of dimensions, reminiscent of scalars (0D), vectors (1D), and matrices (2D). Larger-dimensional tensors are used for advanced information representations.
- Operations: Tensors could be added, multiplied, and remodeled. These operations are extensions of matrix operations to increased dimensions.
- Purposes: Tensors are utilized in varied functions, together with picture and video processing, the place they will symbolize multidimensional information.