Scalars:
A scalar is only a single quantity. It’s the easiest type of knowledge, 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 determine every particular person quantity by index. Usually we give vector daring lowercase identify. Ingredient of vector with italic typeface, in v 1st ingredient as v₁ 2nd ingredient as v₂ & so on. To point the kind of numbers within the vector {ℝ, ℕ, ℤ, and so forth.} 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 house with every ingredient giving coordinate alongside completely different axis.
Properties
- Magnitude and Path
- 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 numerous functions.
Matrices:
A Matrices is 2-D array of quantity recognized by two indices as an alternative of only one. They’re utilized in numerous fields, together with pc graphics, the place they assist in transformations and rotations, and in linear algebra for fixing programs of equations. We normally give matrices uppercase daring typeface. corresponding to A, however To point the kind of numbers & peak(3) and width(3), we are saying like “ A ∈ ℕ³*³ ” & ingredient 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 can be utilized in fixing linear equations and in transformations.
Tensor:
Tensors generalize the ideas of scalars, vectors, and matrices to greater dimensions. A tensor is basically 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 knowledge in one-dimensional to n-dimensional areas, extending the ideas of vectors and matrices.
Properties
- Multi-Dimensional: Tensors can have a number of dimensions, corresponding to scalars (0D), vectors (1D), and matrices (2D). Greater-dimensional tensors are used for complicated knowledge representations.
- Operations: Tensors might be added, multiplied, and remodeled. These operations are extensions of matrix operations to greater dimensions.
- Functions: Tensors are utilized in numerous functions, together with picture and video processing, the place they’ll characterize multidimensional knowledge.