How to represent linear systems as matrix-vector equations

Refer: MATH115 - Linear Algebra, page 133

• We begin with a system of linear equations.
• Then write down the respective coefficient matrix $A$.
• And the solution vector $\vec{b}$.
• We can write each each column of our coefficient matrix as a vector ${\vec{a}}_1,{\vec{a}}_2,{\vec{a}}_3$
• Rewriting our system in these new terms we can see that the system is consistent iff:

$$\vec{b}=x_1{\vec{a}}_1+x_2{\vec{a}}_2+x_3{\vec{a}}_3$$

• Now to represent this “operation” concisely we define the Matrix-Vector Product