Linear Algebra

Résumé du document

Notes from Linear Algebra Chapter 1: Vectors and Matrices
You will learn about vectors, bases vectors and vectors property. You will also learn about matrices, what are matrices in a geometrical aspect and some matrices property.

Extraits

[...] b b′ Linear Transformation Lin. Transformations scale the area of the square bounded by ab, a′ b′ . The area will stretch or shrink based on the Lin. Transformation LA-1-Vectors+Matrices 1 The factor by which the area will scales is actually the determinant of the matrix. Formula : a × b′ − a′ × b Theorem : Det(M1 M2 ) = Det(M1 ).Det(M2 ) For the 3D Lin. Transformation the volume will scale instead of the area. [...]

[...] Properties and Definitions : Square matrices : Square matrices are all the matrices that can be written as Mn×n Basically they have the same number of rows & columns Identity matrices : Identity matrices are square matrices which have only's 1 in the diagonal and zero everywhere else Example : I 1 ] —> It is usually noted as I 1 Trace of a matrices : The trace of a matrices is the sum of all the elements that are on the diagonal. Example : A 3 ] , trace = 3+4 = Invert of a matrix : A matrix is invertible if it determinant =  Notation : A2 ∈ M A−1 is the invert matrix of A Geometric representation : Remember that a 2x2 matrix is the lin. transformation of the bases vector. Its inverse is the lin. transformation that undo the first transformation. [...]

[...] These numbers are called coordinates x x ⎡ ⎤ Notation : Vectors are written in column matrices form → [ ] , y y ⎣ ⎦ z Vectors are also represented by straight line in a n dimensional space with n the numbers of the vector's coordinates. Operation : Vectors can be added, subtracted and multiplied by a scalar. Base vectors Definition : Bases vectors are vectors which will rules over all the other vectors. Causes : All vectors can be written as a linear combination of the bases vectors. u = ai + bj Matrices Geometrical Definition : A 2x2 matrix represents the new bases vectors after a linear transformation a a′ [ ] where each columns represent a transformed base vector. [...]