MAT1503 (LINEAR ALGEBRA) QUESTIONS AND ANSWERS A+ GRADED. Buy Quality Materials!
matrix
A rectangular array of numbers
linear equation in n (j) unknowns
a₁x₁ + a₂x₂ + ... + ax = b
linear equation
x + 3y = 7 or x₁ − 2x₂ − 3x₃ + x₄ = -1 (no products or roots of variables...
matrix
A rectangular array of numbers
linear equation in n (j) unknowns
a₁x₁ + a₂x₂ + ... + aⱼxⱼ = b
linear equation
x + 3y = 7 or x₁ − 2x₂ − 3x₃ + x₄ = -1 (no products or roots of variables)
system of linear equations (linear system)
A finite set of linear equations
solution of a linear equation
A sequence of numbers for which the substitution with variables will make the equation
a true statement
homogeneous linear equations
x₁ − 2x₂ − 3x₃ + x₄ = 0
solution of a linear system
The element is a solution of each equation
solution set (general solution)
All solutions of a linear system with the number sequence as the elements
ordered n-tuple
A linear solution written as (a₁, a₂, ... , aⱼ)
ordered pair
ordered n-tuple if n = 2
ordered triple
ordered n-tuple if n = 3
consistent system
A linear system that has at least one solution
equivalent systems
Two systems of equations that have the same solution set
inconsistent system
A linear system that has no solutions
parameter
An assigned arbitrary value where the linear system has infinite solutions
parametric equations
The solution expressed by the equations using parameters
algebraic operations
1) Add a multiple of one equation to another
2) Multiply an equation by a nonzero constant
3) Interchange two equations
Elementary Row Operations
1) Add a multiple of one row to another row
2) Multiply any row by a nonzero constant
3) Interchange two rows
, augmented matrix
An abbreviation of a linear system in a rectangular array of numbers
elementary matrix
A matrix that was (or could be) produced by performing a single Elementary Row
Operation on an identity matrix
identity matrix
A square matrix with 1's on the main diagonal and zeros everywhere else. Note A×I = A
and I×A = A
Row Echelon Form
A matrix that has leading ones on the main diagonal and zeros below the leading ones.
Reduced Row Echelon Form
A matrix that has leading ones on the main diagonal and zeros above and below the
leading ones.
leading variables
The variables corresponding to the leading 1's in the augmented matrix
free variables
The variables that can be assigned an arbitrary value
Gaussian Elimination
1) Put the matrix in augmented matrix form
2) Use row operations to put the matrix in echelon form
3) Write the equations from the echelon form matrix
4) Solve the equations.
Gauss-Jordan Elimination
1) Put the matrix in augmented matrix form
2) Use row operations to put the matrix in reduced echelon form
3) Write the equations from the echelon form matrix
4) Solve the equations.
trivial solution
The solutions of the homogeneous linear systems are 0
non-trivial solution
The solutions of the homogeneous linear systems are infinite (free variables are used)
Free Variable Theorem for Homogeneous Systems
If a homogeneous linear system has n unknowns, and its augmented matrix has r
nonzero rows in reduced row echelon form, then the system has n - r free variables
entries
The numbers in the matrix
Theorem 1.2.2
A homogeneous linear system with more unknowns than equations has infinitely many
solutions
back-substitution
1) Solve the equations for the leading variables
2) Substitute each equation into all equations above it, starting at the bottom
3) Assign arbitrary values to any free variables
column vector
A matrix with only one column
row vector
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