3.7. Linear combination of vectors#
(Linear combination of vectors)
Let \(\vec{v},\vec{u}_1,\dots,\vec{u}_m\in\mathbb{R}^n\) such that
for some \(\alpha_1,\alpha_2,\dots,\alpha_m\in \mathbb{R}\) [1]. Such a sum is called a linear combination of vectors.
For example
and so we have expressed \((2,0,7)^\mathsf{T}\) as a linear combination of the vectors \((1,5,-1)^\mathsf{T}\) and \((0,-10,9)^\mathsf{T}\). To write a vector \(\vec{v}\) as a linear combination of vectors \(\vec{u}_1, \vec{u}_2, \ldots, \vec{u}_m\) we need to calculate the values of the coefficients \(\alpha_1, \alpha_2, \ldots, \alpha_m\) from equation (3.6). To do this we need to solve the system of linear equations
which can be written as the augmented matrix
and the values of the coefficients \(\alpha_1, \alpha_2, \ldots, \alpha_m\) can be calculated using Gaussian elimination.
Express the vector \(\vec{v} = (7, -2, -11)^\mathsf{T}\) as a linear combination of the vectors
Solution
We need to find the values of the coefficients \(\alpha_1\), \(\alpha_2\) and \(\alpha_3\) in the following linear combination
so we need the solution to the linear system
Using Gauss-Jordan elimination
so \(\alpha_1 = 1\), \(\alpha_2 = -2\) and \(\alpha_3 = 2\). Therefore \(\vec{v}\) can be expressed as the linear combination of vectors
3.7.1. Basis vectors#
A special type of vector is a basis vector which all other vectors in the space can be represented as a linear combination of the basis vectors (we will cover basis in more detail later). In a Cartesian space the simplest basis vectors are unit vectors that point in the co-ordinate directions. In \(\mathbb{R}^3\) we use the basis vectors \(\vec{i} = (1, 0, 0)^\mathsf{T}\), \(\vec{j} = (0, 1, 0)^\mathsf{T}\) and \(\vec{k} = (0, 0, 1)^\mathsf{T}\) (Fig. 3.11).
Using basis vectors we can represent any vector, \(\vec{a} = (a_1, a_2, a_3)^\mathsf{T}\) say, as a linear combination of \(\vec{i}\), \(\vec{j}\) and \(\vec{k}\), i.e.,
For example