MATH 133 Ch1 - Sys of linear eq

What equation describes a line in the plane ?

ax + by = c

What are the 3 types of possible solutions of a sys of linear equations ?

• consistent sys :

  • unique solution

  • infinitely many solutions

• inconsistent sys :

  • no solution

What is the geometric interpretation of a sys of 2 linear equations in 2 variables ?

• either 2 parallel lines (no sol)

• 2 lines that cross in 1 point (1 sol)

• twice the same line (infinitely many solutions)

What is the geometric interpretation of a sys of 2 linear equations in 3 variables ?

• either 2 parallel planes (no sol)

• 2 planes that cross in 1 line (infinitely many solutions)

• twice the same plane (infinitely many solutions)

What are the 3 elementary row operations ?

°the operations that you can perform to obtain a sys that is equivalent

(equivalent = having the same solutions)

• interchanging 2 equations

• mutliply an equation by a non zero number

• add a multiple of an equation to another

This operations can be performed on the system or the augmented matrix.

• interchanging the rows

• mutliply a row by a non zero number

• add a multiple of a row to another

Two methods to solve a simple sys and a sys more complexe ?

• substittion

• elementary operations

What conditions a matrix in row echelon form should satisfy ?

1. All zero rows are at the bottom of the matrix

2. The first non zero entry from the left in efvery row is a 1 (the leading 1)

3. The leading 1 of every row is to the right of any leadind above it

What conditions a matrix should satisfy to be in RREF ?

Reduced Row Echelon Form

• the conditions of an REF matrix

• + each leading 1 is the only non zero entry in its column

If in a matrix, all the entries in a row of the coefficient matrix are zero, and the entry of this rox in the constant matrix is not zero, then ..?

Then the sys is inconsistent

Wha&t is the simplest form of matrix to read the solution ? the solutions ?

RREF, REF

Is there different REF ? different RREF ?

yes, no it’s unique

What is a gaussian elimination ? the gaussian algorithme ?

the operations that lead to reduce the matrix toward REF or REF

What is the rank of a matrix ?

the number of leading ones in any of its REF, or RREF

Suppose the matrix has m rows and n columns, what about the rank possibilities ?

• At most 1 leading 1 per column, so r <= n

• At most 1 leading 1 per row, so r <= m

• » rank <= min(m,n)

Suppose the sys has

• m equations

• n variables

• is consistent

What about the rank r ?

• The sol of the sys has n-r parameters

• If r<n, the sys has infinitely many solutions

• If r=n, the sys has a unique solution (full rank)

What is an homogeneous sys ?

A sys is homogeneous if all the constants (on the RHS) are zero

What is the particularity of an homogeneous sys ?

It always has the trivial solution

x1=01, x2=02 … xn=0n

Can you have an homogeneous and inconsistent sys ?

No

If an homogeneous sys has more variables than equations, then what about the solution ?

It has a non-trivial solution (= has infinitely many solutions)

Is the converse of “If an homogeneous sys has more variables than equations, then it has a non-trivial solution (= has infinitely many solutions)” true ?

No, you can have a homogeneous sys w infinetly many soltions & m=n

What means linear combination ?

We can add mutiples of columns to each other (like we did with rows)

What is a basic solution ?

The basic solution of a matrix only have number in its entries (no paramaeters like s or t)

Any solution of a homogeneous sys is a linear combination of ..?

Is it possible for a non homogeneous sys ?

Any solution of a homogeneous sys is a linear combination of basic solutions

No

Is there basic solutions for non homogeneous sys ?

No

Is the trivial solution a linear combination of basic solution ?

Any solution of a homogeneous sys is a linear combination of basic solutions

For the trivial solution, all parameters = 0

Suppose we have an homogeneous sys in n variables & of rank r

Then ?

• there are n-r basic solutions (one for every of the n-r parameters)

• every solutions is a linear combiantion of the basic solutions

Is the statement “If the nber of col is different than the nber of rows, the sys has infinitely many solutions” True or False ?

False, expl 3 equations 2 variables ended in REF : x=2 ; x=3

Is the statement “If the sys has more variables than equations, then it has infinitely many solutions” True or False ?

True

For an homogeneous sys, does the statement “if n>r, then the sys has infintely many solutions” True or False ?

False

expl : Sys of 3 equations in 2 variables w 1=3, 1=2, 0=0

Is the statement “Given a sys of lin eq with coef matrix A of size mxn, and the correspondant augmented matrix M. If rank(A)<rank(M), then the sys has no solution” True or false ?

True because means at least one row is “000|x”, which means impossible so no solution