Ch.1.1 - Systems of Linear Equations

A system of linear equations has?

(i) no solution, or (ii) exactly one solution, or (iii) infinitely many solutions

What are the elementary row operations?

1. (Replacement) Replace one row by the sum of itself and a multiple of another row. "Add to one row a multiple of another" 2. (Interchange) Interchange two rows 3. (Scaling) multiply all entries in a row by a nonzero constant.

What are the two fundamental questions about a linear system?

1. Is the system consistent; that is, does at least one solution exist? 2. If a solution exists, is it the only one; that is, is the solution unique?

What is a solution of the linear system?

A list (S1, S2, ....., Sn) of numbers that makes each equations a true statement when the values S1,...,Sn are substituted for X1,...,Xn, respectively.

What is a "coefficient matrix" of the system?

A rectangular array, containing the coefficient data of the linear systems. x1 - 2X2 + X3 = 0 2X2 - 8x3 = 8 -4X + 5x2 + 9x3 = - 9 ⎡1 -2 1⎤ ⎢0 2 -8⎢ ⎣-4 5 9⎦

What is an "augmented matrix" of the system?

An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations. x1 - 2X2 + X3 = 0 2X2 - 8x3 = 8 -4X + 5x2 + 9x3 = - 9 ⎡1 -2 1 0⎤ ⎢0 2 -8 8⎢ ⎣-4 5 9 -9⎦

What is a linear equation?

An equation where the variables do not have a power that is equal to one. They will generally be of the form: a1X1 + a2X2 + .... + anXn = b Where a1,.....,an are real or complex numbers, usually known in advance. Th subscript n may be any positive integer.

What is the elimination procedure?

Eliminating one of the variables within the linear system of equations by multiplying the variable by a coefficient and then adding or subtracting to one of the other linear equations. Division may be used as well.

How do you find the solution set of a system of two linear equations in two variables?

Find the intersection of the two lines.

What does the size of the matrix tell you?

How many rows and columns it has. This one has "3x4" ⎡1 -2 1 0 ⎤ ⎢0 2 -8 8 ⎢ ⎣-4 5 9 -9⎦

When does a pair of numbers (X1, X2) satisfy both equations?

I.f.f. the point (X1,X2) lies on both line 1 and line 2.

What does a linear system with no solution look like?

It has a built in contradiction. 0 = 5/2, which is never true, therefore there is no solution.

Why is a system of linear equations (or a linear system)?

It is a collection of one or more linear equations involving the same variables (X1, ...., Xn). e.g. 2X1 - X2 + 1.5X3 = 8 X1 - 0X2 - 4X3 = -7

What is a matrix?

It is a rectangular Array, containing the data of the linear systems.

What is an issue when having a computer do linear algebra calculations?

It only has a finite number of digits, and thus introduces error via the form of truncations, i.e. rounding of the decimal places. This seldom causes problems.

When are two matrices called "row equivalent"?

Only if there is sequence of elementary row operations that transforms one matrix into the other.

What are the three basic operations used to "simplifiy" a linear system?

Replacing one equation by the sum of itself and a multiple of another equation, interchange two equations, and multiply all the terms in an equation by a nonzero constant.

What does "inconsistent" mean in linear algebra terminology?

That a system of linear equations has no solution.

What does "consistent" mean in linear algebra terminology?

That a system of linear equations has one solution or infinitely many solutions.

What is a solution set of a linear solution?

The set of all possible solutions.

If the augmented matrices of two linear systems are row equivalent...?

Then the two systems have the same solution set.

What is important about row operations?

They must be reversible.

What is a good way to check each step?

To do the opposite operation of the one just done and to see if it will yield the previous line.

What does interchange mean?

To swap around i.e row 1 goes in row 2's spot and then row 2 goes in row 1's spot.

When are two linear systems called equivalent?

When they have the same solution set.