Concept Check 1

What is a linear equation?  What is a system of linear equations?  ax+by=c is a linear equation with x and y as variables.  When more than x and y, a linear equation is    a1x1 + a2x2 + . . .+anxn = b where x1, x2, xn are variables.  A finite collection of linear equations in the variables x1, x2,  . . ., xn is called a system of linear equations.  What is a solution for a system of linear equations?  What does it mean to solve a linear system?  Given a linear equation a1x1 + a2x2 + . . . + anxn = b, a sequence s1, s2, . . ., sn of n numbers is called a solution to the equation if a1s1 + a2s2 + . . . ansn = b.  That is, if the equation is satisfied when the substitutions x1=s1, x2=s2, . . ., xn=sn are made.  A sequence of numbers is called a solution to a system of equations if it is a solution to every equation in the system.  When you solve a linear system, you find the number solutions for the several variables in multiple equations that makes them all true.  What are elementary operations, and what effect do they have on the set of solutions for a linear system?  Elementary operations can be routinely performed on systems of linear equations to produce equivalent systems.  The three elementary operations are (I) – Interchange two equations, (II)- Multiply one equation by a nonzero number, (III) – Add a multiple of one equation to a different equation.  The solutions for the linear system do not change; rather, this is a way to make them more clear.  What is the difference between the row-echelon form and the reduced row-echelon form of a matrix?  A matrix is said to be in row-echelon form (and will be called a row-echelon matrix) if it satisfies the following three conditions:  1.All zero rows (consisting entirely of zeros) are at the bottom  2. The first nonzero entry from the left in each nonzero row is a 1, called the leading 1 for that row  3. Each leading 1 is to the right of all leading 1’s in the rows above it.  A row-echelon matrix is said to be in reduced row-echelon form (and will be called a reduced row-echelon matrix) if, in addition, it satisfies the following condition:  4.  Each leading 1 is the only nonzero entry in its column.  What is the difference between leading variables and free variables in a system of linear equations?  To solve a linear system, the augmented matrix is carried to reduced row-echelon form, and the variables corresponding to the leading ones are called leading variables.  It is customary to call the nonleading variables “free” variables, and to label them by new variables s,t,…, called parameters.  What is the Gaussian algorithm? Gaussian elimination?  The Gaussian Algorithm is as follows: Step 1 – If the matrix consists entirely of zeros, stop – it is already in row-echelon form.  Step 2 – Otherwise, find the first column from the left containing a nonzero entry (call it a), and move the row containing that entry to the top position.  Step 3 – Now multiply the new top row by 1/a to create a leading 1.  Step 4 – By subtracting multiples of that row from rows below it, make each entry below the leading 1 zero.  This completes the first row, and all further row operations are carried out on the remaining rows.  Step 5 – Repeat steps 1-4 on the matrix consisting of the remaining rows.  The process stops when either no rows remain at step 5 or the remaining rows consist entirely of zeros.  Gaussian elimination is as follows: To solve a system of linear equations proceed as follows: 1. Carry the augmented matrix to a reduced row-echelon matrix using elementary row operations.  2. If a row [0 0 0  . . . 0 1] occurs, the system is inconsistent.  3. Otherwise, assign the nonleading variables (if any) as parameters, and use the equations corresponding to the reduced row-echelon matrix to solve for the leading variables in terms of the parameters.  

What is the rank of a matrix?  What does the rank of an augmented matrix tell us about the number of solutions to the corresponding linear system?  The rank of matrix A is the number of leading 1’s in any row-echelon matrix to which A can be carried by row operations.  Suppose a system of m equations in n variables is consistent, and that the rank of the augmented matrix is r.  1.The set of solutions involves exactly n-r parameters.  2. If r<n, the system has infinitely many solutions.  3. If r=n, the system has a unique solution.  What does it mean to say that a linear system is homogeneous?  A system of equations in the variables x1,x2,. . ., xn is called homogeneous if all the constant terms are zero – that is, if each equation of the system has the form a1x1+a2x2+ . . .+anxn = 0.  What is the difference between trivial and nontrivial solutions?  Clearly x1=0, x2=0,  . . ., xn=0 is a solution to such a system; it is called the trivial solution.  Any solution in which at least one variable has a nonzero value is called a nontrivial solution.  What is a linear combination of columns (vectors)?  The sum of scalar multiples of several columns is called a linear combination of these columns.  If x=  [                ] and y=[               ] then x+y=[                                  ] and kx=[                        ].  For example, sx+ty is a linear combination of

x and y for any choice of numbers s and t.  What are the basic solutions to the homogeneous linear system?  The gaussian algorithm systematically produces solutions to any homogenous linear system, called basic solutions, one for every parameter.  These are sets of actual numbers.  What is the relationship between the basic solutions and the solution set for a homogenous linear system?  The basic solutions are a part of a homogenous’s solution set, however, not the only components.  The solution set also contains the trivial solution and the infinite linear combinations of the basic solutions.