Decomposition of a vector by basis

In this publication, we will consider how a vector can be decomposed into two basis vectors, and we will also analyze an example of solving a problem on this topic.

Vector decomposition principle

In order to decompose a vector b by basis vectors a₁,…, a_n, it is required to determine such coefficients x₁,…, X_n, for which the linear combination of vectors a₁,…, a_n equals vector b, i.e:

x₁a₁ + … + x_na_n = b

where x₁,…, X_n are vector coordinates b in base a₁,…, a_n

Example of a problem

Let’s decompose the vector b = {16; 1} by two basis vectors m = {2; 1} и n = {1; -3}.

Decision:

1. The vector equation looks like this:

xm + Yn = b

2. Let’s represent it in the form:

3. Now you need to solve the system. From the second equation we get:

x = 1 + 3y.

We substitute the resulting expression into the first equation:

2 · (1 + 3y) + y = 16

2 + 6y + y = 16

7y = 14

y = 2

Consequently, the x = 1 + 3y = 1 + 2 · 2 = 7.

Answer: b = 7m + 2n.