Linear Algebra Vectors: definition, operations. Systems of vectors: linear dependence and independence, basis
Vectors and systems of vectors Vectors: definitions Vectors: Zero vector : if all the coordinates of a vector are zero, we call the vector a zero vector and denote it by θ. Identity vectors: a vector in which some coordinate is 1 while all the remaining coordinate are 0. Equality: Two vectors u and v are said to be equal if they are of the same size and their corresponding coordinates are equal. Vector Coordinates Zero vector Θ=(0, 0, …, 0) Identity vectors in 4 -space e 1=(1, 0, 0) e 2=(0, 0, 0) e 3=(1, 0, 0) e 4=(1, 0, 0) Equality u=v
Vectors and systems of vectors Operations Scalar multiplication of a vector: If u is a vector in n –space and c is real number, then the product c. A is a vector in n –space , obtained by multiplying each coordinates of u by the constant c. Addition and subtraction of vectors: Two vectors in n –space u and v can be added (or subtracted) by adding (or subtracting) their corresponding coordinates. Inner product: let u and v vectors in n –space , then the inner product uv is obtained by calculating the product of corresponding coordinate in u and v and then finding the sum of all n of these. Norm of vector: Scalar multiplication of a vector Addition and subtraction of vectors Inner product of two vectors
Vectors and system of vectors Properties Commutative law for addition of vectors: u+v=v+u Associative law for addition of vectors : (u+v)+w=u+(v+w) Distributive law for linear operations: c(u+v)=cu+cv Distributive law for inner product: u(v+w)=uv+uw, (u+v)w=uw+vw Zero vector: Let u is a vector in n-space and 0 is the zero vector in n-space, then u+0=u. Commutative law for inner product of vectors: u∙v=v∙u : Triangular inequality:
Vectors and system of vectors Linear dependence and linear independence
Скачать презентацию Linear Algebra Vectors definition operations Systems of vectors
vectors.pptx