# Decomposition of the vector in the basis

To 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 - example

Example 1. Decompose the vector b = {8; 1} by basis vectors p = {1; 2} and q = {3; 1}.

**Solution:** Form the vector equation:

which can be written as a system of linear equations

1x + 3y = 8 | |

2x + 1y = 1 |

from the first equation express x

x = 8 - 3y | |

2x + y = 1 |

Substitute x in the second equation

x = 8 - 3y | |

2(8 - 3y) + y = 1 |

x = 8 - 3y | |

16 - 6y + y = 1 |

x = 8 - 3y | |

5y = 15 |

x = 8 - 3y | |

y = 3 |

x = 8 - 3·3 | |

y = 3 |

x = -1 | |

y = 3 |

**Answer:** b = -p + 3q.

Vectors
Vectors Definition. Main information
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Length of a vector
Direction cosines of a vector
Equal vectors
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Angle between two vectors
Vector projection
Addition and subtraction of vectors
Scalar-vector 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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