1: Systems of Linear Equations- Algebra
- Page ID
- 70181
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Solve a system of linear equations algebraically in parametric form.
This chapter is devoted to the algebraic study of systems of linear equations and their solutions. We will learn a systematic way of solving equations of the form
\[\left\{\begin{array}{rrrrrrrrrrrrr} 3x_1 &+& 4x_2 &+& 10x_3 &+& 19x_4 &-& 2x_5 &-& 3x_6 &=& 141\\ 7x_1 &+& 2x_2 &-& 13x_3 &-& 7x_4 &+& 21x_5 &+& 8x_6 &=& 2567\\ -x_1 &+& 9x_2 &+& \frac 32x_3 &+& x_4 &+& 14x_5 &+& 27x_6 &=& 26\\ \frac 12x_1 &+& 4x_2 &+& 10x_3 &+& 11x_4 &+& 2x_5 &+& x_6 &=& -15 \end{array}\right. \nonumber\]
In Section 1.1, we will introduce systems of linear equations, the class of equations whose study forms the subject of linear algebra. In Section 1.2, will present a procedure, called row reduction, for finding all solutions of a system of linear equations. In Section 1.3, you will see hnow to express all solutions of a system of linear equations in a unique way using the parametric form of the general solution.
- 1.1: Systems of Linear Equations
- This page introduces \(\mathbb{R}^n\) as the set of ordered \(n\)-tuples of real numbers for labeling geometric points, focusing on linear equations' structure, consistency, and solutions. It discusses the geometric interpretation of solutions in n-dimensional space, illustrating how linear equations define lines or planes.
- 1.2: Row Reduction
- This page introduces the elimination method for solving systems of linear equations using augmented matrices and row operations. It defines applicable operations, including scaling, replacement, and swapping, and discusses achieving row echelon and reduced row echelon forms. The process includes identifying unique, inconsistent, and infinite solutions through examples, emphasizing the significance of pivot positions.
- 1.3: Parametric Form
- This page explains parametric form and free variables in solving linear equations. It outlines how to express solution sets using free variables, demonstrating the infinite solutions available when at least one variable is free. The text also classifies systems of linear equations based on their augmented matrix forms, identifying three scenarios: inconsistent systems with no solutions, unique solutions, and systems with infinitely many solutions due to free variables.