Differential Equations and Linear Algebra, 4th Edition

4.3 Linear Combinations and Independence of Vectors

In Example 5 of Section 4.2 we solved the homogeneous linear system

$$\begin{array}{rrrrrrrrr}{x}_1& +& 3x_2& -& 15x_3& +& 7x_4& =& 0\\ x_1& +& 4x_2& -& 19x_3& +& 10x_4& =& 0\\ 2x_1& +& 5x_2& -& 26x_3& +& 11x_4& =& 0.\end{array}$$ (1)

We found that its solution space W consists of all those vectors x in ${\mathbf{\text{R}}}^4$ that have the form

$$\mathbf{x}=s(3,4,1,0)+t(2,-3,0,1).$$ (2)

We therefore can visualize W as the plane in ${\mathbf{\text{R}}}^4$ determined by the vectors ${\mathbf{\text{v}}}_1=(3,4,1,0)$ and ${\mathbf{\text{v}}}_2=(2,-3,0,1)$. The fact that every solution vector is a combination [as in (2)] of the particular solution vectors ${\mathbf{\text{v}}}_1$ and ${\mathbf{\text{v}}}_2$ gives us a tangible understanding of the solution space W of the system in (1).

More generally, we know from Theorem 2 in Section 4.2 that the solution set V of any $m\times n$ homogeneous linear system $\mathbf{\text{Ax}}=\mathbf{0}$ is a subspace of ${\mathbf{\text{R}}}^n$. In order to ...