Chapter 1 - Systems of Equations

Question 1

What defines a linear equation?

Answer 1

Any equation that can be written as:

a1x1 + a2x2 + a3x3 + … + anxn = b

Question 2

Theorem 1.1.1:

If a sequence of elementary operations is performed on a system of linear equations, the resulting system has [______] solutions as the original, and so the two systems are equivalent.

Answer 2

If a sequence of elementary operations is performed on a system of linear equations, the resulting system has the same solutions as the original, and so the two systems are equivalent.

Question 3

State the three elementary row operations.

Answer 3

1. Interchange two rows.
2. Multiply one row by a non-zero number.
3. Add a multiple of one row to a different row.

Question 4

State the inverses of the three elementary row operations.

Answer 4

1. Interchange the rows again.
2. Multiply the row by 1/k (k is the non-zero number).
3. Add the negative of the multiple to the new row.

Question 5

Define ‘row-echelon form’.

Answer 5

A matrix is in row-echelon form if the following conditions are satisfied:

a) All zero rows are at the bottom.
b) The first non-zero entry from the left in each non-zero row is a 1, called the leading 1 for that row.
c) Each leading 1 is to the right of all leading 1s above it.

Question 6

Define ‘reduced row-echelon form’.

Answer 6

A matrix is in reduced row-echelon form if the following conditions are satisfied:

a) All zero rows are at the bottom.
b) The first non-zero entry from the left in each non-zero row is a 1, called the leading 1 for that row.
c) Each leading 1 is to the right of all leading 1s above it.
d) Each leading 1 is the only non-zero entry in its column.

Question 7

Theorem 1.2.1:

Every matrix can be brought to reduced row-echelon form by a sequence of [_________]

Answer 7

Theorem 1.2.1:

Every matrix can be brought to reduced row-echelon form by a sequence of elementary row operations.

Question 8

State the Gaussian Algorithm.

Answer 8

1. If the matrix consists of 0s, stop.
2. Otherwise, find the first column from the left containing a non-zero entry (k) and move the row containing that entry to the top.
3. Multiply the new top row by 1/k.
4. Subtract multiples of the top row from all rows below it in order to make all entries below the leading 1 zero.
5. Repeat steps 1-4 for the rest of the matrix rows.

Question 9

Define the rank of a matrix.

Answer 9

The rank of a matrix is the number of leading 1s in any row-echelon matrix to which A can be carried by elementary row operations.

Question 10

Theorem 1.2.2

Suppose a system of m equations in n variables is consistent, and the rank of the matrix is r.

1. The set of solutions involves [____] parameters.
2. If [____], the system has infinitely many solutions.
3. If [____], the system has a unique solution.

Answer 10

Theorem 1.2.2

Suppose a system of m equations in n variables is consistent, and the rank of the matrix is r.

1. The set of solutions involves n-r parameters.
2. If r < n, the system has infinitely many solutions.
3. If r = n, the system has a unique solution.

Question 11

Explain what is meant by a homogeneous system of equations.

Answer 11

All the constant terms are 0:

ax + by + cz + … = 0
dx + ey + fz + … = 0
…

Question 12

State the trivial solution of a homogeneous system of n equations in n variables.

Answer 12

x1 = 0, x2 = 0, x3 = 0, …, xn = 0

Question 13

Explain what is meant by a non-trivial solution.

Answer 13

Any solution containing at least one non-zero value for a variable.

Question 14

Theorem 1.3.1

If a homogeneous system of the linear equation has more variables than equations, then it has [______] of [______] solutions.

Answer 14

If a homogeneous system of the linear equation has more variables than equations, then it has an infinite number of non-trivial solutions.

Question 15

Theorem 1.3.2

Let A be an m x n matrix of rank r, and consider the homogenous system in n variables with A as coefficient matrix.

Then,

1) The solution has [______] basic solutions.
2) Every solution is a [______] of these basic solutions.

Answer 15

Theorem 1.3.2

Let A be an m x n matrix of rank r, and consider the homogenous system in n variables with A as coefficient matrix.

Then,

1) The solution has n - r basic solutions.
2) Every solution is a linear combination of these basic solutions.

Question 16

State the ‘Junction Rule’.

Answer 16

At a junction in a network, the total flow into that junction must equal the total flow out.