Elementary Linear Algebra
Rachelle Kun
Quick research tells me that 'Linear Algebra' is more complicated than I would think at a glance. Do I actually need to revisit calculus? What other specific subjects will prepare me to handle university-level math again? I'm willing to get text books and tutors, as I have nearly a year to prepare.
Some ability to do mathematical reasoning will also matter. How does one acquire that? Here unfortunately I'm not sure which books to recommend. This is mostly stuff I learned in 8th through 11th grades, and partly in various undergraduate math courses that were not primarily concerned with how mathematical reasoning is done.
I personally didn't find my first course in linear algebra very hard at all. It was very algorithmic so I didn't have to think too hard or be too creative to solve any of the problems. But that doesn't mean that you will have the same experience. I know there were other people in my class who thought it was difficult.
As for prereqs, you probably won't need calculus. At the very most I could see your professor including Wronskians which will require you to be able to differentiate a function. But it'll probably be functions which differentiate easily, like polynomials and trig functions. What you will need is a pretty solid understanding of high school algebra and a little bit of geometry -- mostly analytic geometry (using coordinates) rather than synthetic geometry (using just the axioms).
It's easy to see (in hindsight!) that the cross product can't be quite right...this is because it's only defined in 3-dimensions; which unlike the inner product is defined for all spaces; and in fact, in more advanced treatments it's replaced by the wedge product.
The other distinction that's nots usefully emphasised is duality; that is the notion of a form; forms are dual to vectors, and admit all the same operations; and in fact, taking the dual of a form just gives you back the vectors.
This is useful since forms appears later in calculus on manifolds: ie differential geometry; and in physics - for example, using the the technology of forms the four maxwell equations are reduced to two; and also in that form we can ask them to hold not just in out 3d Euclidean space, but on any manifold of arbitrary shape and dimension.
Taking a course in multivariate calculus helped a lot with some concepts in my opinion, but it wasn't required. I haven't used linear algebra in any subsequent courses yet, unless you count some of the theory we studied doing proofs (which had nothing to do with LA).
MATH 2200 - Elementary Linear Algebra(3) Credit Hours
Basic concepts and techniques of linear algebra; introduces systems of linear equations, matrices, determinants, geometric vectors, vector spaces, general real vector spaces, linear transformations, eigenvalues and eigenvectors, orthogonality, and inner product spaces. Fall, spring, summer semesters. Lecture 3 hours. Prerequisites: MATH 1950 or MATH 2030 with minimum grades of C or department head approval.
So given a square matrix $M\in M_n\times n(L)$, it is left invertible if and only if it is right invertible. What are the known relations between the left kernel of $M$ in $L^n$ and its right kernel ?
A different and inequivalent question is why the dimension of the kernel of a linear transformation is unchanged by extension of scalars (for a map $f : R\to S$ of division rings). The reason is that a division ring will always be flat as a module over a division ring.
Then consider the skew field $\mathbbQ((a,b))$ (Malcev Neumann series, for example as in arXiv:math/0405133) which is the set of functions $\Gamma \rightarrow \mathbbQ$ with well-ordered supports (and usual operations). Then, the matrix $$\beginpmatrixba & a\\b^2 & b\endpmatrix$$has its columns left proportional but not right proportional. So, the vector space generated by the columns on the left has dimension one and on the right has dimension 2. In fact $M$ is (two-sided) invertible. One has $$M^-1=\beginpmatrix[b,a]^-1 & [a,b]^-1ab^-1\\-b[b,a]^-1 & -b[a,b]^-1ab^-1+b^-1\endpmatrix$$and $M$ is a (two-sided) a zero divisor for the opposite field $K^op$Concerning question 1 using elementary operations, one can prove
Now, you have the non-degenerate pairing (still by matrix multiplication)$$\langle\ \ \rangle\ :\ K^1\times n\otimes_K K^n\times 1\rightarrow K^1\times 1\simeq K$$ (this time, the two spaces are considered as $K-K$-bimodules).
Continuation of single-variable differential and integral calculus. Topics covered include: inverse and hyperbolic functions; techniques of integration; polar and parametric equations; infinite sequences, series, power series and Taylor series; applications of integration. Primarily for mathematics, physical science and engineering majors. Prerequisite: MATH 1 with a minimum grade of C. 90 hours lecture. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A; C-ID# MATH 221, MATH 900 S (if taken with MATH 1).
Introduction to diff erential equations including the conditions under which a unique solution exists, techniques for obtaining solutions, and applications. Techniques include generation of series solutions, use of Laplace Transforms, and the use of eigenvalues to solve linear systems. Generation of exact solutions, approximate solutions, and graphs of solutions using MATLAB. Prerequisite: MATH 3 with a minimum grade of C. 54 hours lecture, 27 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A; C-ID# MATH 240.
Designed for majors in mathematics and computer science, this course provides an introduction to discrete mathematical structures used in Computer Science and their applications. Course content includes: Propositional and predicate logic; rules of inference; quantifiers; elements of integer number theory; set theory; methods of proof; induction; combinatorics and discrete probability; functions and relations; recursive definitions and recurrence relations; elements of graph theory and trees. Applications include: analysis of algorithms, Boolean algebras and digital logic circuits. Students who have completed, or are enrolled in, CS 17 may not receive credit. Prerequisite: MATH 1 with a minimum grade of C (May be taken concurrently), CS 1 with a minimum grade of C (May be taken concurrently). 72 hours lecture, 18 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A; C-ID# COMP 152.
This course focuses on the development of quantitative reasoning skills through in-depth, integrated explorations of topics in mathematics, including real number systems and subsystems. Emphasis is on comprehension and analysis of mathematical concepts and applications of logical reasoning. Prerequisite: MATH 50 with a minimum grade of C or MATH 55 with a minimum grade of C or NMAT 255 with a minimum grade of C or NMAT 250 with a minimum grade of C. 54 hours lecture. AA/AS GE. Transfer: CSU, UC.
College algebra core concepts relating to Science, Technology, Engineering and Mathematics (STEM) and Business fields are explored, such as: polynomial, rational, radical, exponential, absolute value, and logarithmic functions; systems of equations; theory of polynomial equations; and analytic geometry. Multiple representations, applications and modeling with functions are emphasized throughout. May not receive credit if Mathematics 20 or 45 have been completed. Prerequisite: MATH 55 with a minimum grade of C or MATH 55B with a minimum grade of C or NMAT 255 with a minimum grade of C. 72 hours lecture, 18 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU GE: B4, IGETC: 2A; C-ID# MATH 151.
Linear functions, systems of linear equations and inequalities, exponential and logarithmic functions and applications, matrices, linear programming, mathematics of finance, sets and Venn diagrams, combinatorial techniques and an introduction to probability. Applications in business, economics and social sciences. Prerequisite: MATH 50 with a minimum grade of C or MATH 55 with a minimum grade of C or MATH 55B with a minimum grade of C or NMAT 250 with a minimum grade of C or NMAT 255 with a minimum grade of C. 72 hours lecture. AA/AS GE. Transfer: CSU, UC*; CSU GE: B4; IGETC: 2A; C-ID# MATH 130. * MATH 1, 33, and 34 combined: maximum UC credit, one course.
Functions and their graphs; limits of functions; differential and integral calculus of algebraic, exponential and logarithmic functions. Applications in business, economics, and social sciences and use of graphing calculators. Partial derivatives and the method of LaGrange multipliers. Prerequisite: MATH 55 with a minimum grade of C or MATH 55B with a minimum grade of C or NMAT 255 with a minimum grade of C. 90 hours lecture. AA/AS GE. Transfer: CSU, UC*; CSU GE: B4; IGETC: 2A; C-ID# MATH 140. * MATH 1, 33, and 34 combined: maximum UC credit, one course.
Trigonometry includes definitions of the trigonometric functions and their inverses, graphs of the trigonometric functions and their inverses, trigonometric equations, trigonometric expressions and identities, including proofs, an introduction to vectors, polar coordinates and complex numbers. Applications include solving right triangles and solving triangles using the law of sines and the law of cosines. Prerequisite: MATH 55B with a minimum grade of C or MATH 55 with a minimum grade of C or NMAT 255 with a minimum grade of C. 72 hours lecture, 18 hours laboratory. AA/AS GE. Transfer: CSU; CSU GE: B4; C-ID# MATH 851.
Descriptive statistics, including measures of central tendency, dispersion and position; elements of probability; confi dence intervals; hypothesis tests; two-population comparisons; correlation and regression; goodness of fit; analysis of variance; applications in various fields. Introduction to the use of a computer software package to complete both descriptive and inferential statistics problems. Prerequisite: MATH 55 with a minimum grade of C or MATH 55B with a minimum grade of C or MATH 50 with a minimum grade of C or NMAT 250 with a minimum grade of C or NMAT 255 with a minimum grade of C. 72 hours lecture, 18 hours laboratory. AA/AS GE. Transfer: CSU, UC; CSU GE: B4; IGETC: 2A; C-ID# MATH 110.