Solving Systems of Linear Equations

Question 1

What is the central core of linear algebra?

Answer 1

To solve systems of linear

equations (the unknowns are only multiplied by numbers).

Question 2

Define a linear system.

Answer 2

A linear system is a collection of m equations in n variables.

Question 3

What is the goal of linear algebra?

Answer 3

To solve the linear system simultaneously.

Question 4

What is the general solution of a linear system?

Answer 4

The set of all solutions.

Question 5

What is an inconsistent linear system?

Answer 5

A linear system that has no solution.

Question 6

What is a consistent linear system?

Answer 6

A linear system that has at least one solution.

Question 7

What is a homogeneous linear system?

Answer 7

A linear system in which all the constants on the

right-hand-side are zeros.

Question 8

What is an inhomogeneous linear system?

Answer 8

A linear system in which at least one of the constants on the
right-hand-side is non-zero.

Question 9

What kind of linear systems are always consistent?

Answer 9

Homogeneous linear systems.

Question 10

What is a degenerate linear system?

Answer 10

A linear system where all the coefficients are zeros.
0x1+0x2+···0xn = b, b∈R.
If b≠0, this equation has no solution and if b=0, every
x = (x1,x2,···,xn)∈R^n
is a solution

Question 11

What are the three elementary row operations?

Answer 11

1. Add a multiple of one equation to another
2. Exchange equations
3. Multiply an equation by a non-zero scalar.
   (All three can be reversed)

Question 12

What is an augmented matrix?

Answer 12

A matrix composed of a coefficient matrix A and a column of constants b.
[A|b]

Question 13

If there are m equations and n unknowns in a system, what is the size of the coefficient matrix A?

Answer 13

mxn (m rows and n columns)

Question 14

When is a matrix said to be in Row Echelon Form

(REF)?

Answer 14

1. All the zero rows are at the bottom.
2. The first non-zero entry in each row is a 1, which is called a leading 1 or a pivot.
3. Each leading 1 is to the right of the one in the row above.

Question 15

When is a matrix said to be in Reduced Row Echelon Form

(RREF)?

Answer 15

If, in addition to being in REF, each leading 1 is the only non-zero entry in its column.

Question 16

What are 4 remarks about REF’s and RREF’s?

Answer 16

1 From REF of the augmented matrix of a linear system we will be able to tell if the system is inconsistent or has a unique solution, or has infinitely many solutions.

2. From the RREF we can read off the solution.
3. Variables corresponding to columns which do not have a leading 1 are called non-leading variables and these are the parameters in the general solution.
4. Non-leading variables are responsible for a linear system to have infinitely many solutions.

Question 17

When are two matrices A and B are row equivalent?

Answer 17

Two matrices A and B are row equivalent (A∼B or B∼A) if B can be obtained from A by a finite sequence of elementary row operations.

Question 18

How may RREF’s does any matrix have?

Answer 18

A unique one.

Question 19

What are the 3 rules for reading off the type of general solution from REF?

Answer 19

1. If the augmented matrix (in REF) contains a row like
   [0, 0, · · · , 0|b] where b≠0, then it is inconsistent and there is no solution,
2. If every column of REF has a leading 1, then there is a unique solution,
3. If there is a column in a REF that does not have a leading 1 (which corresponds to a non-leading variable), then there are infinitely many solutions.

Question 20

What is the procedure for writing down the general solution from a consistent RREF?

Answer 20

1. If there is a unique solution (i.e. every column has a leading 1), then the solution is the column of constants.
2. Otherwise, identify the leading and non-leading variables in the RREF. Then,
3. 1 Write down the equation for each row and solve for the leading variables in terms of non-leading variables.
4. 2 Set each non-leading variable to a different parameter and substitute them into the equations for the leading variables.
5. 3 Write down the general solution, including all variables.

Question 21

What are the 5 steps of the Gaussian Elimination Algorithm for reducing systems to REF?

Answer 21

1. If M is the zero matrix, stop!
2. Find the left-most non-zero column, and interchange the top-row with another (if necessary) to bring a non-zero entry to the first row of this column.
3. Scale the first row, as necessary, to get a leading 1.
4. Add suitable multiple of the top row to the other rows to create zeros below the leading 1.
5. This completes the operations with the first row. Keep first
   row and go back to step 1.

When the algorithm stops, the resulting matrix will be in REF.

Question 22

What are the 3 additional steps of the Gaussian Elimination Algorithm for reducing systems to RREF?

Answer 22

6. If the right-most leading 1 is in row 1, stop!
7. Start from the right-most leading 1 and add suitable multiples of that row to create zeros above that leading 1.
8. Cover up the row we used and go step (6).

When the algorithm stops, the resulting matrix would be in RREF.

Question 23

What is a matrix?

Answer 23

A a rectangular array of numbers or entries.