Linear algebra explained in four pagesExcerpt from the NO BULLSHIT GUIDE TO LINEAR ALGEBRAby Ivan SavovAbstract—This document will review the fundamental ideas of linear algebra.We will learn about matrices, matrix operations, linear transformations anddiscuss both the theoretical and computational aspects of linear algebra. Thetools of linear algebra open the gateway to the study of more advancedmathematics. A lot ofknowledge buzzawaits you if you choose to follow thepath ofunderstanding, instead of trying to memorize a bunch of formulas.I. INTRODUCTIONLinear algebra is the math of vectors and matrices. Letnbe a positiveinteger and letRdenote the set of real numbers, thenRnis the set of alln-tuples of real numbers. A vector~v∈Rnis ann-tuple of real numbers.The notation “∈S” is read “element ofS.” For example, consider a vectorthat has three components:~v= (v1, v2, v3)∈(R,R,R)≡R3.A matrixA∈Rm×nis a rectangular array of real numbers withmrowsandncolumns. For example, a3×2matrix looks like this:A=24a11a12a21a22a31a3235∈24RRRRRR35≡R3×2.The purpose of this document is to introduce you to the mathematicaloperations that we can perform on vectors and matrices and to give you afeel of the power of linear algebra. Many problems in science, business,and technology can be described in terms of vectors and matrices so it isimportant that you understand how to work with these.PrerequisitesThe only prerequisite for this tutorial is a basic understanding of high schoolmath concepts1like numbers, variables, equations, and the fundamentalarithmetic operations on real numbers: addition (denoted+), subtraction(denoted-), multiplication (denoted implicitly), and division (fractions).You should also be familiar withfunctionsthat take real numbers asinputs and give real numbers as outputs,f:R→R. Recall that, bydefinition, theinverse functionf-1undoesthe effect off. If you aregivenf(x)and you want to findx, you can use the inverse function asfollows:f-1(f(x)) =x. For example, the functionf(x) = ln(x)has theinversef-1(x) =ex, and the inverse ofg(x) =√xisg-1(x) =x2.II. DEFINITIONSA. Vector operationsWe now define the math operations for vectors. The operations we canperform on vectors~u= (u1, u2, u3)and~v= (v1, v2, v3)are: addition,subtraction, scaling, norm (length), dot product, and cross product:~u+~v= (u1+v1, u2+v2, u3+v3)~u-~v= (u1-v1, u2-v2, u3-v3)α~u= (αu1, αu2, αu3)||~u||=qu21+u22+u23~u·~v=u1v1+u2v2+u3v3~u×~v=(u2v3-u3v2, u3v1-u1v3, u1v2-u2v1)The dot product and the cross product of two vectors can also be describedin terms of the angleθbetween the two vectors. The formula for the dotproduct of the vectors is~u·~v=k~ukk~vkcosθ. We say two vectors~uand~vareorthogonalif the angle between them is90◦. The dot product oforthogonal vectors is zero:~u·~v=k~ukk~vkcos(90◦) = 0.