Mathematics 2, Professor M. Dall’Aglio
Linear Algebra
General Recap : Week 1
A transformation through a function as the one above implies for f to
associate every element of A to a unique element of B.
A Matrix is an array of real numbers in a rectangular
fashion with M rows and N columns.
M is the set of all matrices that have m rows and n columns.
From a linear system we can derive the associated matrix, which can
either be augmented (A|b) or simple (A).
{
A linear system is said to be in Echelon form if the associated matrix
has the following form:
Where the * are known as “Pivots”, non zero numbers whose relation
with the matrix “steps” determine the number of solutions.
• 1 solution : whenever every step = 1 pivot
• 0 solutions : whenever the last row has, all zeros except the result
(if A|b) or all zeros (if A).
• ∞ solutions : whenever 1 step ≠ 1 pivots but also • (=parametric eq.)
, The Gauss Reduction process is a process that allows every matrix to
be turned into a matrix in echelon form, and it implies the following
simplification methods:
1. Exchanging row orders
2. Multiplying one equation for a scalar λ
3. Exchange one equations with “itself + another row”
By combining 1, 2 and 3 any linear system can be transformed.
The Rank of a Matrix:
Given a matrix A in echelon form (either originally or transformed in
such) the rank of A is the “number of non zero rows”
Vectors :
Vectors are ordered n-tuples of real numbers.
The set containing every n-tuple is
Operations on vectors:
Given the vectors u = (u1, u2, u2) v = (v1, v2, v3)
Linear Algebra
General Recap : Week 1
A transformation through a function as the one above implies for f to
associate every element of A to a unique element of B.
A Matrix is an array of real numbers in a rectangular
fashion with M rows and N columns.
M is the set of all matrices that have m rows and n columns.
From a linear system we can derive the associated matrix, which can
either be augmented (A|b) or simple (A).
{
A linear system is said to be in Echelon form if the associated matrix
has the following form:
Where the * are known as “Pivots”, non zero numbers whose relation
with the matrix “steps” determine the number of solutions.
• 1 solution : whenever every step = 1 pivot
• 0 solutions : whenever the last row has, all zeros except the result
(if A|b) or all zeros (if A).
• ∞ solutions : whenever 1 step ≠ 1 pivots but also • (=parametric eq.)
, The Gauss Reduction process is a process that allows every matrix to
be turned into a matrix in echelon form, and it implies the following
simplification methods:
1. Exchanging row orders
2. Multiplying one equation for a scalar λ
3. Exchange one equations with “itself + another row”
By combining 1, 2 and 3 any linear system can be transformed.
The Rank of a Matrix:
Given a matrix A in echelon form (either originally or transformed in
such) the rank of A is the “number of non zero rows”
Vectors :
Vectors are ordered n-tuples of real numbers.
The set containing every n-tuple is
Operations on vectors:
Given the vectors u = (u1, u2, u2) v = (v1, v2, v3)
