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Decomposition of the vector in the basis

To decomposition, the vector b on the basis vectors a1, ..., an, you must find the coefficients of x1, ..., xn, for which a linear combination of vectors a1, ..., an is equal to vector b:

x1a1 + ... + xnan = b,

the coefficients x1, ..., xn are called the coordinates of the vector b in the basis a1, ..., an.

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:

xp + yq = b,

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.

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