Question

Write the solution set of the given homogeneous system in parametric vector form

x₁ +3x₂ −5x₃ = 0
x₁ +4x₂ −8x₃ = 0
−3x₁ −7x₂+ 9x₃ = 0

Answer 1

To find the solution set of the system of equations in parametric vector form, you perform row reduction on the system's matrix to find out that x₃ can be a free parameter. Then, express x₁ and x₂ in terms of x₃, and write the solution set with x₃ as the parameter.

Step-by-step explanation:

To find the solution set of the given homogeneous system of equations in parametric vector form, we need to solve the system:

• x₁ +3x₂ −5x₃ = 0
• x₁ +4x₂ −8x₃ = 0
• −3x₁ −7x₂+ 9x₃ = 0

Let's express the system in matrix form and perform row reduction to echelon form.

After simplifying, you might end up with a matrix that looks something like this:

[1 0 a
0 1 b
0 0 0]

This suggests that x₃ (the third variable) can be a free parameter, for example, let x₃ = t.

Now, we can express x₁ and x₂ in terms of t using the first two equations:

x₁ = -at
x₂ = -bt

Thus, in parametric vector form, the solution set is:

t(-a, -b, 1)

where R is the set of all real numbers.