Question

Let \({{\bf{a}}_1}\) \({{\bf{a}}_2}\), and b be the vectors in \({\mathbb{R}^{\bf{2}}}\) shown in the figure, and let \(A = \left( {\begin{aligned}{*{20}{c}}{{{\bf{a}}_1}}&{{{\bf{a}}_2}}\end{aligned}} \right)\). Does the equation \(A{\bf{x}} = {\bf{b}}\) have a solution? If so, is the solution unique? Explain.

Short answer

The system of equations \(A{\bf{x}} = {\bf{b}}\) has a unique solution.

Step-by-step solution

Construct the graph with a grid

Consider the figure shown below:

On the \({x_1}{x_2}\)-plane, the lines between \({{\bf{a}}_1}\) and the origin and between \({{\bf{a}}_2}\) and the origin form a grid. Each point can be defined by using the grid.

Determine the solution

In the above figure,move some steps in the direction of the vectors to reach towards vectors\({{\bf{a}}_1}\),\({{\bf{a}}_2}\), and bfrom the origin.

There is always a unique way to reach these vectors. It means,\(A{\bf{x}} = {\bf{b}}\)has a solution.

Thus, the system of equations \(A{\bf{x}} = {\bf{b}}\) has a unique solution.