Big Idea 1: Describing and Understanding Solutions PDF
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This document outlines learning targets for linear algebra and includes concepts like finding solutions to systems of linear equations and understanding the relationships between homogeneous and non-homogeneous systems. It focuses on describing solutions using matrix operations and vector representations.
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Big Idea 1: Describing and understanding solutions. =================================================== The first big idea in our linear algebra courses is that you can *describe* and *understand* all the solutions of a system of linear equations, or prove that no solution exists. Reread this idea...
Big Idea 1: Describing and understanding solutions. =================================================== The first big idea in our linear algebra courses is that you can *describe* and *understand* all the solutions of a system of linear equations, or prove that no solution exists. Reread this idea and think about what this means, and what you have already learned. To develop this idea we propose three learning targets below. Following each learning target is a series of \"I can \...\" statements which are meant to help you get to this target. 1\. Find the solutions of a system of linear equations or prove that no solution exists. i. I can bring a matrix to reduced row echelon form. ii. I can say if a system of equations has 0, 1 or infinitely many solutions. iii. When a system has at least one solution, I can write one solution down. iv. When a system has more than one solution, I can describe the solutions using free variables. 2\. Learn a language to effectively organize the solutions of a system of linear equations. i. I can translate a system of linear equations to a vector equation, and vice versa. ii. I can translate a system of linear equations to a matrix equation of the form *A***x** = **b**. iii. I can describe in complete English sentences the phrases \"span\", \"linear combination\", \"linear independence\" and \"linear dependence\". iv. I can determine if a set of vectors is linearly independent or not. v. I can determine if a given vector is in the span of some other given vectors. vi. I can determine if a collection of vectors spans **R*^n^***. 3\. Understand the relationship between the solutions of a homogeneous and the related non-homogeneous system of equations. i. I can state the content of Theorem 6 in Chapter 1, Section 5 and use it to solve a non-homogeneous system of equations. ii. I know how to interpret the geometry behind Theorem 6 in Chapter 1, Section 5.
