# The Fundamental Theorem of Linear Algebra

# The Fundamental Theorem of Linear Algebra

This chapter summarizes pictorially some of the linear algebraic foundations discussed thus far by revisiting the fundamental theorem of linear algebra, the unifying view of matrices, vectors, and their interactions. To make the discussion helpful and informal yet rigorous, and to complement the slightly more formal introduction of the basic ideas given in an earlier chapter, here the theorem’s pictorial representation is emphasized. The discussions cover the forward problem, when **A** ɛ ℝM×N maps an **x** ɛ ℝN from **A**’s domain onto **b** ɛ ℝM in **A**’s range, how **A** transforms **x** into **b**; and the inverse problem, discussed in detail in chapter 9, section 9.4.1.

*Keywords:*
linear algebra, matrices, vectors, forward problem, inverse problem

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