Linear Algebra

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This book helps students to master the material of a standard undergraduate linear algebra course. The material is standard in that the topics covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. The audience is also standard: sophomores or juni

Systems of linear equations are common in science and mathematics. These two examples from high school science (O'Nan 1990) give a sense of how they arise. The first example is from Physics. Suppose that we are given three objects, one with a mass known to be 2 kg, and are asked to find the unknown

Definition 1.1 A **linear equation** in variables ${x}_{1} , {x}_{2} , … , {x}_{n}$ has the form ${a}_{1} {x}_{1} + {a}_{2} {x}_{2} + {a}_{3} {x}_{3} + ⋯ + {a}_{n} {x}_{n} = d$ where the numbers ${a}_{1} , … , {a}_{n} ∈ ℝ$ are the equation's **coefficients** and $d ∈ ℝ$ is the **constant**. An n-tup

A linear system with a unique solution has a solution set with one element. A linear system with no solution has a solution set that is empty. In these cases the solution set is easy to describe. Solution sets are a challenge to describe only when they contain many elements.

$\{ {{( \begin{matrix}0 \\ 4 \\ 0 \\ 0 \\ 0\end{matrix} )}_{⏟}}_{\begin{matrix} \\ \text{particular} \\ \text{solution}\end{matrix}} + {{w ( \begin{matrix}1 \\ − 1 \\ 3 \\ 1 \\ 0\end{matrix} ) + u ( \begin{matrix}1 / 2 \\ − 1 \\ 1 / 2 \\ 0 \\ 1\end{matrix} )}_{⏟}}_{\begin{matrix} \\ \text{unrestrict

This subsection is optional. Later material will not require the work here. ## Comparing Set Descriptions A set can be described in many different ways. Here are two different descriptions of a single set: $\{ ( \begin{matrix}1 \\ 2 \\ 3\end{matrix} ) z \ | \ z ∈ ℝ \} \ \text{and} \ \{ ( \begin{matr

This is a PASCAL routine to do $k {ρ}_{i} + {ρ}_{j}$ to an augmented matrix. PROCEDURE Pivot(VAR LinSys : AugMat; k : REAL; i, j : INTEGER) VAR Col : INTEGER; BEGIN FOR Col:=1 TO NumVars+1 DO LinSys[j,Col]:=k*LinSys[i,Col]+LinSys[j,Col]; END; Of course this is only one part of a whole program, but i

For readers who have seen the elements of vectors before, in calculus or physics, this section is an optional review. However, later work will refer to this material so it is not optional if it is not a review. In the first section, we had to do a bit of work to show that there are only three types

"Higher-dimensional geometry" sounds exotic. It is exotic— interesting and eye-opening. But it isn't distant or unreachable. We begin by defining one-dimensional space to be the set ${ℝ}^{1}$.

We've translated the first section's results about solution sets into geometric terms for insight into how those sets look. But we must watch out not to be mislead by our own terms; labeling subsets of ${ℝ}^{k}$ of the forms $\{ \overset{→}{p} + t \overset{→}{v} \ | \ t ∈ ℝ \}$ and $\{ \overset{→}{p

After developing the mechanics of Gauss' method, we observed that it can be done in more than one way. One example is that we sometimes have to swap rows and there can be more than one row to choose from. Another example is that from this matrix $( \begin{matrix}2 & 2 \\ 4 & 3\end{matrix} )$ Gauss'

Gaussian elimination coupled with back-substitution solves linear systems, but it's not the only method possible. Here is an extension of Gauss' method that has some advantages. Example 1.1 To solve $\begin{matrix}x & + & y & − & 2 z & = & − 2 \\ & & y & + & 3 z & = & 7 \\ x & & & − & z & = & − 1\en

We will close this section and this chapter by proving that every matrix is row equivalent to one and only one reduced echelon form matrix. The ideas that appear here will reappear, and be further developed, in the next chapter. The underlying theme here is that one way to understand a mathematical

The linear systems in this chapter are small enough that their solution by hand is easy. But large systems are easiest, and safest, to do on a computer. There are special purpose programs such as LINPACK for this job. Another popular tool is a general purpose computer algebra system, including both

An economy is an immensely complicated network of interdependences. Changes in one part can ripple out to affect other parts. Economists have struggled to be able to describe, and to make predictions about, such a complicated object. Mathematical models using systems of linear equations have emerged

Gauss' method lends itself nicely to computerization. The code below illustrates. It operates on an $n \ × \ n$ matrix a, pivoting with the first row, then with the second row, etc. for (pivot_row = 1; pivot_row <= n - 1; pivot_row++) { for (row_below = pivot_row + 1; row_below <= n; row_below++) {

The diagram below shows some of a car's electrical network. The battery is on the left, drawn as stacked line segments. The wires are drawn as lines, shown straight and with sharp right angles for neatness. Each light is a circle enclosing a loop.

We are using Gauss' Method to solve the linear systems in this book because it is easy to understand, easily shown to give the right answers, and fast. It is fast in that, in all the by-hand calculations we have needed, we have gotten the answers in only a few steps, taking only a few minutes. Howev

The first chapter began by introducing Gauss' method and finished with a fair understanding, keyed on the Linear Combination Lemma, of how it finds the solution set of a linear system. Gauss' method systematically takes linear combinations of the rows. With that insight, we now move to a general stu

Definition 1.1 A **vector space** (over $ℝ$) consists of a set V along with two operations "$+$" and "$⋅$" subject to these conditions. - For any $\overset{→}{v} , \overset{→}{w} ∈ V : \overset{→}{v} + \overset{→}{w} ∈ V$ . - For any $\overset{→}{v} , \overset{→}{w} ∈ V : \overset{→}{v} + \overset{→

One of the examples that led us to introduce the idea of a vector space was the solution set of a homogeneous system. For instance, we've seen in [Example 1.4]( such a space that is a planar subset of ${ℝ}^{3}$. There, the vector space ${ℝ}^{3}$ contains inside it another vector space, the plane. De

The prior section shows that a vector space can be understood as an unrestricted linear combination of some of its elements— that is, as a span. For example, the space of linear polynomials $\{ a + b x \ | \ a , b ∈ ℝ \}$ is spanned by the set $\{ 1 , x \}$. The prior section also showed that a spac

For the right to left implication assume that $\overset{→}{v} ∈ [ S ]$ to show that $[ S ] = [ S ∪ \{ \overset{→}{v} \} ]$ by mutual inclusion. The inclusion $[ S ] ⊆ [ S ∪ \{ \overset{→}{v} \} ]$ is obvious. For the other inclusion $[ S ] ⊇ [ S ∪ \{ \overset{→}{v} \} ]$, write an element of $[ S ∪