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Appendix B Perspectives

In this appendix we draw attention to the many ways in which a single idea can be viewed from multiple perspectives.

Figure B.0.1. All the Same Story

Given an \(m\times n\) matrix \(A\text{,}\) with columns \(A_j,\ j=1,\ldots,n\text{,}\) and vectors \(\vec{b}\in \mathbb{R^m}\) and \(\vec{x}\in \mathbb{R^n}\) we get the following equivalences:

The Linear System

Given the Set of Vectors

The Matrix Equation

Given the Linear Transformation

\(\left(\sum_{j=1}^{n} a_{ij}x_i\right)=b_i\) is




Consistent for some \(\vec{b}\text{,}\) but the solution is not unique

\(\vec{b}\in span\left(\mathcal{B}_A\right)\)

Has infinitely many solutions

\(\vec{b}\in Image\left(T_A\right)\)

Consistent for some \(\vec{b}\) and the solution is unique

They are linearly independent and \(\vec{b}\in span\left(\mathcal{B}_A\right)\)

Has exactly one solution, \(n \lt m\text{,}\) \(A\) has a pivot in every column

\(\vec{b}\in Image\left(T_A\right)\text{,}\) and \(T_A\) is 1-1

Consistent for all \(\vec{b}\text{,}\) but the solution is not unique


Has infinitely many solutions, \(A\) has a pivot in every row, \(m \lt n\)

\(Image\left(T_A\right)=\mathbb{R}^m\text{,}\) \(T_A\) is onto

Consistent for all \(\vec{b}\) and the solution is unique

\(\mathcal{B}_A\) is a basis for \(\mathbb{R}^m\)

Has exactly one solution, \(A\) is invertible, \(m=n\)

\(Image\left(T_A\right)=\mathbb{R}^m\text{,}\) \(T_A\) is 1-1 and onto


\(\vec{b}\not\in span\left(\mathcal{B}_A\right)\)

Has no solution

\(\vec{b}\not\in Image\left(T_A\right)\)

Table B.0.2. True From a Certain Point of View