
Decomposition of the vector in the basisTo decomposition, the vector b on the basis vectors a_{1}, ..., a_{n}, you must find the coefficients of x_{1}, ..., x_{n}, for which a linear combination of vectors a_{1}, ..., a_{n} is equal to vector b: x_{1}a_{1} + ... + x_{n}a_{n} = b, the coefficients x_{1}, ..., x_{n} are called the coordinates of the vector b in the basis a_{1}, ..., a_{n}. Decomposition of the vector in the basis  exampleExample 1. Decompose the vector b = {8; 1} by basis vectors p = {1; 2} and q = {3; 1}.
Solution: Form the vector equation: xp + yq = b,which can be written as a system of linear equations
from the first equation express x
Substitute x in the second equation
Answer: b = p + 3q.
Vectors
Vectors Definition. Main information
Component form of a vector with initial point and terminal point
Length of a vector
Direction cosines of a vector
Equal vectors
Orthogonal vectors
Collinear vectors
Coplanar vectors
Angle between two vectors
Vector projection
Addition and subtraction of vectors
Scalarvector multiplication
Dot product of two vectors
Cross product of two vectors (vector product)
Scalar triple product (mixed product)
Linearly dependent and linearly independent vectors
Decomposition of the vector in the basis
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