Topic 8: Systems of linear equations

Question 1

How can we solve 3x + y = 5, 2x + 2y = 4 with matrix notation (3, 3)

Answer 1

-A = (TL TR BL BR) (3 1, 2 2)
-x = (T B) (x, y)
-b = (T B) (5, 4)

-A^-1 = (1/2, -1/4, -1/2, 3/4)
-x^* = (1/2, -1/4, -1/2, 3/4)(5, 4)
-x* = (3/2, 1/2)

Question 2

What is cramers rule (4)

Answer 2

-The system of n equations and n unknowns has a unique solution iff A is nonsingular
-x_i = D_i/|A|
-D_i is the determinant of the Augmented matrix Ā, formed by replacing the i^th column of A with the RHS constants B
-Cramers rule avoids the computation of A^-1

Question 3

How can we solve 2x + y = 1, x - y + z = 0, 2y - z = 3 with cramers rule (1,4,3)

Answer 3

• |A| = (TL->TR, ML->MR, BL->BR) |2 1 0, 1 -1 1, 0 2 -1| = -1 ≠ 0

-To find D₁, replace column 2, 1, 0, with 1, 0, 3 and work the determinant out, and the same for D₂ and D₃
-D₁ = 2, -D₂ = -5,
-D₃ = -7

-X = D₁/|A| = -2
-Y = D₂/|A| = 5
-Z = D₃/|A| = 7

Question 4

What is a solution of a linear system of equations known as (2)

Answer 4

-A solution of a linear system of equations is an assignment of values to the variables x₁, …, x_n such that each of the equations is satisfied
-The set of all possible solutions is called the solution set

Question 5

What are the number of solutions for a linear system (3, 3)

Answer 5

A linear system can either have
-an infinite amount of solutions
-1 solution
0 solutions

For 2 variable systems, the solution set is either:
-A line (infinite)
-A single point (1)
-An empty set (0)

Question 6

When is a system of equations consistent (1)

Answer 6

-If it has at least one solution

Question 7

When are the rows of a coefficient matrix linearly independent/dependent (2)

Answer 7

-The rows of a coefficient matrix are linearly independent if none of the rows can be derived algebraically from the others
-If a row can be derived from other rows of the system, these rows are linearly dependent

Question 8

When is an equation redundant in a system of linear equations (1)

Answer 8

-If no info is lost by removing it

Question 9

What is the importance of all rows of the coefficient matrix being linearly independent (1)

Answer 9

-All the rows of the coefficient matrix being linearly independent is a necessary and sufficient condition for the nonsingularity of the corresponding coefficient matrix A, where A is a square matrix

Question 10

What is the rank of a matrix (2)

Answer 10

-If the maximum number of linearly independent rows/columns that can be found in a matrix is r, the matrix is said to be of rank r
-The rank of (mxn) matrix can be at most min{m, n}