Question

Use the reduced coefficient matrix to write a general solution in terms of one or more column matrices. also determine the dimension of the solution space and a basis for this space. 6x₁ - x₂ + x = 0

x₁ - x₄ + x₅ = 0
x ₁- 2x₅ = 0

Answer 1

To find the general solution in terms of one or more column matrices, we need to use the reduced coefficient matrix. The dimension of the solution space is 1 and a basis for this space can be represented by the column matrix [0 0 0 0 1]T.

Step-by-step explanation:

To find the general solution in terms of one or more column matrices, we need to use the reduced coefficient matrix. Given the system of equations:

6x₁ - x₂ + x₃ = 0

x₁ - x₄ + x₅ = 0

x₁ - 2x₅ = 0

We can rewrite it in matrix form as:

[6 -1 1 0 0][x₁ x₂ x₃ x₄ x₅]T = [0 0 0]

Using Gaussian elimination, we can reduce the coefficient matrix to row-echelon form:

[1 0 0 0 3/2][x₁ x₂ x₃ x₄ x₅]T = [0 0 0]

So the general solution can be expressed as:

x₁ = -3/2x₅

x₂ = 0

x₃ = 0

x₄ = 0

x₅ = x₅

The dimension of the solution space is 1 since there is one free variable x₅.

A basis for this space can be represented by the column matrix [0 0 0 0 1]T.