Decomposition of the vector in the basis

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

x₁a₁ + ... + x_na_n = b,

the coefficients x₁, ..., x_n are called the coordinates of the vector b in the basis a₁, ..., 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:

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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