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2: Systems of Linear Equations- Geometry

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    We have already discussed systems of linear equations and how this is related to matrices. In this chapter we will learn how to write a system of linear equations succinctly as a matrix equation, which looks like \(Ax=b\text{,}\) where \(A\) is an \(m \times n\) matrix, \(b\) is a vector in \(\mathbb{R}^m\) and \(x\) is a variable vector in \(\mathbb{R}^n\). As we will see, this is a powerful perspective. We will study two related questions:

    1. What is the set of solutions to \(Ax=b\text{?}\)
    2. What is the set of \(b\) so that \(Ax=b\) is consistent?

    The first question is the kind you are used to from your first algebra class: what is the set of solutions to \(x^2-1=0\). The second is also something you could have studied in your previous algebra classes: for which \(b\) does \(x^2=b\) have a solution? This question is more subtle at first glance, but you can solve it in the same way as the first question, with the quadratic formula.

    In order to answer the two questions listed above, we will use geometry. This will be analogous to how you used parabolas in order to understand the solutions to a quadratic equation in one variable. Specifically, this chapter is devoted to the geometric study of two objects:

    1. the solution set of a matrix equation \(Ax=b\text{,}\) and
    2. the set of all \(b\) that makes a particular system consistent.

    The second object will be called the column space of \(A\). The two objects are related in a beautiful way by the rank theorem in Section 2.8.

    Instead of parabolas and hyperbolas, our geometric objects are subspaces, such as lines and planes. Our geometric objects will be something like 13-dimensional planes in \(\mathbb{R}^{27}\text{,}\) etc. It is amazing that we can say anything substantive about objects that we cannot directly visualize.

    We will develop a large amount of vocabulary that we will use to describe the above objects: vectors (Section 2.1), spans (Section 2.2), linear independence (Section 2.5), subspaces (Section 2.6), dimension (Section 2.7), coordinate systems (Section 2.8), etc. We will use these concepts to give a precise geometric description of the solution set of any system of equations (Section 2.4). We will also learn how to express systems of equations more simply using matrix equations (Section 2.3).

    • 2.1: Vectors
      This page discusses the fundamental concepts of vectors in \(\mathbb{R}^n\), including their algebraic and geometric interpretations, addition, subtraction, and scalar multiplication. It highlights the parallelogram law, linear combinations of vectors, and their real-world applications, especially in physical quantities like velocity.
    • 2.2: Vector Equations and Spans
      This page examines the relationships between systems of linear equations and vector equations, focusing on the span of vectors and its significance. It defines the span as all linear combinations of given vectors and explains that a vector is in the span if the corresponding linear equation is consistent.
    • 2.3: Matrix Equations
      This page explores the matrix equation \(Ax = b\), defining key concepts like consistency conditions, the relationship between matrix and vector forms, and the significance of spans. It explains that for \(Ax = b\) to have solutions, the vector \(b\) must lie within the span of \(A\)'s columns. Systems have solutions for all \(b\) if \(A\) has a pivot in every row.
    • 2.4: Solution Sets
      This page discusses homogeneous and inhomogeneous linear systems, focusing on equations \(Ax=0\) and \(Ax=b\). It defines homogeneous systems as having zero constants, always including the trivial solution \(x=0\). Solutions are expressed as spans of vectors, with their dimensions linked to free variables. For consistent equations \(Ax=b\), solutions can be formulated as a particular solution plus the homogeneous solution set.
    • 2.5: Linear Independence
      This page covers the concepts of linear independence and dependence among vectors, defining linear independence as having only the trivial zero solution in equations. It outlines criteria for testing independence, including the role of pivot columns in matrices, and explains scenarios where vectors are dependent, such as scalar multiples and inclusion of the zero vector.
    • 2.6: Subspaces
      This page defines subspaces in \(\mathbb{R}^n\) and outlines criteria for a subset to qualify as a subspace, including non-emptiness and closure under addition and scalar multiplication. It offers examples of valid and invalid subspaces, discusses conditions specific to \(\mathbb{R}^2\), and explains spanning sets of subspaces, particularly the column and null spaces of matrices.
    • 2.7: Basis and Dimension
      This page discusses the concept of a basis for subspaces in linear algebra, emphasizing the requirements of linear independence and spanning. It covers the basis theorem, providing examples of finding bases in various dimensions, including specific cases like planes defined by equations. The text explains properties of subspaces such as the column space and null space of matrices, illustrating methods for finding bases and verifying their dimensions.
    • 2.9: The Rank Theorem
      This page explains the rank theorem, which connects a matrix's column space with its null space, asserting that the sum of rank (dimension of the column space) and nullity (dimension of the null space) equals the number of columns. It includes examples demonstrating how different ranks and nullities influence solution options in linear equations, emphasizing the theorem's importance in understanding the relationship between solution freedom and system properties without direct calculations.
    • 2.8: Bases as Coordinate Systems
      This page explains how a basis in a subspace serves as a coordinate system, detailing methods for computing \(\mathcal{B}\)-coordinates and converting to standard coordinates. It illustrates finding a basis through row reduction, using examples to demonstrate the representation of vectors as linear combinations of basis vectors. Visual aids support the explanations, emphasizing the verification of linear independence and span to confirm a basis.

    This page titled 2: Systems of Linear Equations- Geometry is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Dan Margalit & Joseph Rabinoff via source content that was edited to the style and standards of the LibreTexts platform.