1 Linear Equations In Linear Algebra Flashcards
A linear system (a system of linear equations):
A collection of one or more linear equations involving the same variable.
Linear equation:
An equation that can be written in the form a1x1 + a2x2 + … + anxn = b where b and the “a” coefficients are real or complex numbers.
Solution (a solution to the system):
A list of numbers that make each equation a true statement when the values are substituted for their respective x.
Solution set:
The set of all possible solutions.
Two linear systems are considered equivalent if…
They have the same solution set.
The solution set is a representation of…
The intersection point of the graphs of the lines.
A solution set can have how many answers?
None, one, or infinitely many
A linear system is consistent if…
It has one or infinitely many solutions.
A system of linear equations is inconsistent if…
It has no solution.
Matrix notation:
The essential information of a linear system.
What does it mean if a matrix is a coefficient matrix (augmented matrix)?
The matrix includes values of b.
Size of a matrix:
Tells how many rows and columns a matrix has in mxn form, where m is the number of rows and n is the columns
What are the elementary row operations?
Replacement, Interchanging, and Scaling
What is the row operation Replacement?
Replace one row by the sum of itself and a multiple of another row.
What is the row operation Interchanging?
Interchange two rows.
What is the row operation Scaling?
Multiply all entries in a row by a nonzero constant.
What does it mean if two matrices are called row equivalent?
Some series of operations can transform one of the matrices into the other.
What does it mean if the augmented matrices of two systems are row equivalent?
The two systems have the same solution set.
