\documentclass{article} %\pagestyle{empty} \setlength{\topmargin}{-.2in} \setlength{\oddsidemargin}{-.3in} \setlength{\textheight}{8.5in} \setlength{\textwidth}{7.2in} \usepackage{amssymb} \newcommand{\stbar}{\;|\;} \newcommand{\R}{\mathbb{R}} \begin{document} \begin{center} {\Large Some Study Hints} \end{center} I felt like one of the main problems with Test 1 was that many people did not study properly. So, here are some study hints that you might find useful for the future. \begin{enumerate} \item Be familiar with how to do computations and what they mean. For this class, our main tool is Gaussian elimination. So, there are several issues you need to know here: \begin{itemize} \item How to perform Gaussian elimination (this was tested in problem 3). \item How to interpret the results of the Gaussian elimination. This is the most important part of Gaussian elimination. The key point is that Gaussian elimination is always used to solve equations. What the solution tells you is different depending on what you are looking for. (this was tested in problem 1 and 4 where the use of Gaussian elimination was different for each problem). \end{itemize} For example, we have thus far seen Gaussian elimination in the following situations: \begin{enumerate} \item Solving equations and writing down the solutions of those equations. This usually involved writing the solution down in ``parametric form.'' (Section 1.3-1.4) \item Determining if a vector $v$, is in the span of a set of vectors $\{v_1, \ldots, v_n\}$. (This can also be phrased differently: determining if a vector is in the column space of a matrix. Why do these really mean the same thing?) Here the equation that we are solving in this case: \[ c_1 v_1 + \cdots + c_n v_n = v \] Written as a system and translated into an augmented matrix yeilds: \[ \left[ \begin{array}{ccc|c} | & & | & | \\ v_1 & \cdots & v_n & v \\ | & & | & | \end{array} \right] \] which we can perform gaussian elimination on. The results of the elimination tell us information about the unknowns: $\{c_1, \ldots, c_n\}$. Questions to think about: \begin{itemize} \item How can we tell if we have no solutions from the Gaussian eliminaion? What does this mean in in terms of the vectors $v, v_1, \ldots, v_n$? \item How can we tell if we have a single solution? What does this mean in in terms of the vectors $v, v_1, \ldots, v_n$? \item How can we tell if we have an infinite number of solutions? What does this mean in in terms of the vectors $v, v_1, \ldots, v_n$? \end{itemize} \item Determining if a set of vectors is linearly independent of linearly dependent. Here, we have a set of vectors $\{v_1, \ldots, v_n\}$ and we want to solve the equation \[ c_1 v_1 + \cdots + c_n v_n = 0 \] Written as a system and translated into an augmented matrix yeilds: \[ \left[ \begin{array}{ccc|c} | & & | & \vdots \\ v_1 & \cdots & v_n & 0 \\ | & & | & \vdots \end{array} \right] \] which we can perform gaussian elimination on. This is similar to the previous equation, except that this question yeilds a \emph{homogeneous} equation instead of a non-homogeneous equation. The results of the elimination tell us information about the unknowns: $\{c_1, \ldots, c_n\}$. Questions to think about: \begin{itemize} \item Is it possible to have no solutions? Think of this in terms of the equation above and the unknowns $c_1, \ldots, c_n$. \\ (Answer: no, it is not possible.) \item Is it possible to have exactly one solution? Think of this in terms of the equation above and the unknowns $c_1, \ldots, c_n$. What does this mean in in terms of the vectors $v, v_1, \ldots, v_n$? \\ (Answer: it is possible and it means that the vectors are linearly independent.) \item How can we tell if we have an infinite number of solutions? What does this mean in in terms of the vectors $v, v_1, \ldots, v_n$? \\ (Answer: it means the vectors are linearly dependent.) \\ In this case, the reduced matrix should tell us even more. It should tell us about the relationship between the vectors in the set. Without memorizing a bunch of tricks, interpreting this data required thinking about the system of equations the we just solved. It particular the reduced matrix will tell us: \begin{itemize} \item What is a maximal linearly independent set. \item How do the other vectors depend on the vectors in the maximal linearly independent set. \end{itemize} \end{itemize} \end{enumerate} \item Know your theorems! One reason theorems are important is because we use them to prove other results. But, the main reasons theorem are important is because we use them to make our work easier. Here are some of the theorems and how they should be used: \begin{enumerate} \item Theorem 1, page 45: Every matrix is row equivalent to a matrix in echelon form. \begin{itemize} \item This tells us that performing Gaussian elimination will always work (which is nice to know when you are solving equations). \end{itemize} \item More unknowns theorem, page 49: A system of linear equations with more unknowns then equations will either fail to have any solutions or will have an infinite number of solutions. \begin{itemize} \item This is nice because if we have more unknowns then equations, we know that there will not be a single solutions -- there will either be no solutions (inconsistent) or the solution will have to be written in ``parametric form.'' \end{itemize} \item Theorem 1, page 59: A linear system is solvable if and only if the vector of constants belongs to the column space of the coefficient matrix. \begin{itemize} \item This tells us when a system is consistent or inconsistent. In practice, this helps us answer questions such as section 1.3: 6,7,8 \end{itemize} \item Theorem 1, page 87: The pivot columns for a matrix $A$ form a maximal linearly independent set of the column vectors of $A$. The nonpivot columns are linear combinations of the pivot columns. \begin{itemize} \item This helps us ``trim'' down a set of vectors. If we have a set of vectors, $S$, we can put them into a matrix $A$ and perform eliminaion on $A$. By looking at the columns that are pivots, we can extract a maximal linearly independent set from the set $S$. By using this theorem in conjunction with our knowledge about interpreting the results of elimination, we can write the nonpivot vectors as combinations of the pivot vectors. \end{itemize} \end{enumerate} \newpage \item Know your definitions and examples. There are lots of things to keep track of and it will be very difficult to perform well in this (and other!) classes if you don't speak the language of the class. The first step here is making sure you know the definitions and terminology. This doesn't mean memorizing definitions from the book -- it means that you are familiar with the definitions and have worked enough problems to be comfortable with the meaning. Here are some examples: \begin{itemize} \item Vector space (what are the examples?) \item Vector (this is any element of a vector space), \item Linear system (including the way to write a linear system in matrix form.) \item Linear combination. \item Row vector, column vector. \item Linearly independent, linearly dependent sets. \item Span. \item Consistent and inconsistent. \item Homogeneous and non-homogeneous. \item Echelon form, reduced row echelon form. \item Parametric form \item Rank \item Augmented matrix \item Pivots, pivot variables, free variables. \item Column space \item Null space \item Subspace \item Product of a matrix and a column vector. \item Maximal linear independent set. \end{itemize} \end{enumerate} \end{document}