Let's start by setting up the system of equations based on the information given:
1) α + 1/√2 * β = -0.521
2) α + 2√2 * β = -0.585
To solve this system of equations, we can rewrite the equations in matrix form:
[1, 1/√2] [α] = [-0.521]
[1, 2√2] [β] = [-0.585]
Now, this can be solved using a method of linear algebra which is to use matrix algebra. For this, we set up a 2x2 matrix A with coefficients of α and β on the left side of equations and a 2x1 matrix b with the constants Ei and Ej on the right side of the equations.
Matrix A will look like:
| 1 1/√2 |
| 1 2√2 |
Matrix b will look like:
| -0.521 |
| -0.585 |
Now, we can solve for [α, β] using a method of matrix algebra called 'solve' function. This function will solve the system of equations and will give us the values of α and β in the form of a solution vector.
The solution vector is (-0.4996666666666667, -0.030169889330625997)
So, α = -0.4996666666666667 and β = -0.030169889330625997.