3) Solutions of systems of linear equations

Question 1

What are the potential geometric interpretations of systems with 2 equations in 2 variables

Answer 1

• Parallel Lines - no solutions
• Intersect at a point - unique solution
• Same line - infinitely many solutions

Question 2

What are the potential geometric interpretations of systems with 3 equations in 3 variables

Answer 2

• 3 parallel / 2 parallel / Triangular Prism - no solutions
• Intersect at a single point - unqiue solution
• Intersect at a line - solution set is infinite
• Same Planes - solution set is infinite

Question 3

What is a hyperplane

Answer 3

A solution set of an equation in n-dimensional space (R^n)

Question 4

What are the inverses of the row operations

Answer 4

Question 5

What does it mean for A and B to be row equivalent

Answer 5

If a matrix B can be obtained from a matrix A by applying a sequence of elementary row operations to A, then we say that A and B are row equivalent

Question 6

What is a matrix in REF (row echelon form)

Answer 6

• Any rows consisting entirely of zeros are at the bottom
• In each non-zero row, the first non-zero entry (called the leading entry or pivot) is in a column to the left of any leading entries below it

Question 7

Describe examples of REF matrices

Answer 7

Question 8

What is an upper triangular and lower triangular matrix

Answer 8

• A matrix A is upper triangular if Aij = 0 whenever i > j, i.e., every entry below the leading diagonal is zero
• A is lower triangular if Aij = 0 whenever i < j, i.e., every entry above the leading diagonal is zero

Question 9

What is a LU Decomposition

Answer 9

(Where E is the row operations we apply to A to put it in REF)

Question 10

Can we always find a LU Decomposition of a sqaure matrix

Answer 10

• Not all square matrices have an LU decomposition
• However we can always find an LU decomposition after permuting them rows of A, i.e., we can write P A = LU, where P is a permutation matrix (a matrix obtained by permuting the rows of the identity matrix)

Question 11

What is a matrix in RREF (reduced row echelon form)

Answer 11

• It is in REF
• Each pivot is 1
• All entries in a column above a pivot are zero

Question 12

Describe examples of RREF matrices

Answer 12

Question 13

What is Gauss-Jordan elimination

Answer 13

Given a system of linear equations, the process of solving the system
by using row operations to reduce its augmented matrix to RREF, and solving the system of linear equations corresponding to the RREF by back substitution

Question 14

What properties are equivalent to a matrix being invertiable

Answer 14

Question 15

Describe the proof of the properties that are equivalent to a matrix being invertiable (Don’t need to know)

Answer 15

Question 16

If A is invertible how can we find A^-1 using row operations

Answer 16

The elementary row operations which In transform A into In will transform (A|In) into (In|A−1)