Scalars:
A scalar is only a single quantity. It’s the easiest type of information, having solely magnitude with none path. Scalars are sometimes utilized in equations and calculations. Examples of scalars embrace temperature, mass, and velocity. 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 path. We are able to establish every particular person quantity by index. Usually we give vector daring lowercase identify. Ingredient of vector with italic typeface, in v 1st component as v₁ 2nd component as v₂ & so on. To point the kind of numbers within the vector {ℝ, ℕ, ℤ, and many others.} and the dimension of vector, we use notation like ℝⁿ & ℝ³ n/3-Dimensional vector containing actual numbers.
We are able to consider vector as level in n-Dimensional area with every component giving coordinate alongside completely different axis.
Properties
- Magnitude and Course
- Illustration: Vectors might be represented graphically as arrows or numerically as arrays.
- Operations: Vectors might 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 purposes.
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 pc 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. akin to A, however To point the kind of numbers & top(3) and width(3), we are saying like “ A ∈ ℕ³*³ ” & component in italics like A₂₁ or ∱(A)₂₁.
Properties
- Rows and Columns
- Operations: Matrices might be added, subtracted, and multiplied. Matrix multiplication will not be commutative, that 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 larger 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 characterize 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, akin to scalars (0D), vectors (1D), and matrices (2D). Larger-dimensional tensors are used for advanced information representations.
- Operations: Tensors might be added, multiplied, and remodeled. These operations are extensions of matrix operations to larger dimensions.
- Purposes: Tensors are utilized in varied purposes, together with picture and video processing, the place they will characterize multidimensional information.
Conclusion:
