Home

Try the Free Math Solver or Scroll down to Tutorials!

Enter expression, e.g. (x^2-y^2)/(x-y)

Enter expression, e.g. x^2+5x+6

Enter expression, e.g. (x+1)^3

Enter a set of expressions, e.g. ab^2,a^2b

Enter equation to solve, e.g. 2x+3=4

Enter equation to graph, e.g. y=3x^2-1

Depdendent Variable

Number of equations to solve:

Equ. #1:

Equ. #2:

Equ. #3:

Equ. #4:

Equ. #5:

Equ. #6:

Equ. #7:

Equ. #8:

Equ. #9:

Solve for:

Enter inequality to solve, e.g. 2x+3>4

Enter inequality to graph, e.g. y<3x^2-1

Dependent Variable

Number of inequalities to solve:

Ineq. #1:

Ineq. #2:

Ineq. #3:

Ineq. #4:

Ineq. #5:

Ineq. #6:

Ineq. #7:

Ineq. #8:

Ineq. #9:

Solve for:

Math solver on your site

Please use this form if you would like
to have this math solver on your website,
free of charge.

Name:

Email:

Your Website:

Msg:

ON SOLUTIONS OF LINEAR EQUATIONS

Homework Section 1.3: 4,14,34,48,50,26*,46*

MATRIX. A rectangular array of numbers is called a matrix

A matrix with m rows and n columns is called a m × n matrix. A matrix with one column is a column
vector. The entries of a matrix are denoted a_ij , where i is the row number and j is the column number.

ROW AND COLUMN PICTURE. Two interpretations

”Row and Column at Harvard”

Row picture: each b_i is the dot product of a row vector with .
Column picture: is a sum of scaled column vectors .

EXAMPLE. The system of linear equations

is equivalent to where A is a coefficient
matrix and and are vectors.

The augmented matrix (separators for clarity)

In this case, the row vectors of A are

The column vectors are

Row picture:

Column picture:

SOLUTIONS OF LINEAR EQUATIONS. A system

with m equa-
tions and n unknowns is defined by the m × n matrix A and the vector ~b.
The row reduced matrix rref(B) of B determines the number of solutions
of the system Ax = b. The rank rank(A) of a matrix A is the number of
leading ones in rref(A). There are three possibilities:
• Consistent: Exactly one solution. There is a leading 1 in each column
of A but none in the last column of the augmented matrix B.
• Inconsistent: No solutions. There is a leading 1 in the last column of
the augmented matrix B.
• Consistent: Infinitely many solutions. There are columns of A with-
out leading 1.

If rank(A) = n, then there is exactly 1 solution.
If rank(A) < rank(A|b),there are no solutions.
If rank(A) = rank(A|b) < n: there are ∞ solutions.

(exactly one solution)	(no solution)	(infinitely many solutions)

MURPHYS LAW.

”If anything can go wrong, it will go wrong”.
”If you are feeling good, don’t worry, you will get over it!”
”For Gauss-Jordan elimination, the error happens early in
the process and get unnoticed.

MURPHYS LAW IS TRUE. Two equations could contradict each other. Geometrically, the two planes do
not intersect. This is possible if they are parallel. Even without two planes being parallel, it is possible that
there is no intersection between all three of them. It is also possible that not enough equations are at hand or
that there are many solutions. Furthermore, there can be too many equations and the planes do not intersect.

RELEVANCE OF EXCEPTIONAL CASES. There are important applications, where ”unusual” situations
happen: For example in medical tomography, systems of equations appear which are ”ill posed”. In this case
one has to be careful with the method.

The linear equations are then obtained from a method called the Radon
transform. The task for finding a good method had led to a Nobel
prize in Medicis 1979 for Allan Cormack. Cormack had sabbaticals at
Harvard and probably has done part of his work on tomography here.
Tomography helps today for example for cancer treatment.

MATRIX ALGEBRA. Matrices can be added, subtracted if they have the same size:

They can also be scaled by a scalar λ: