For this question, refer to the lecture section on Linear Independence and the supplemental reading. Recall the definitions of dependence and independence:

Dependence means that at least one equation can be algebraically derived from the others (i.e., can be written as a linear combination of other equations). There are an infinite number of solutions that will satisfy the conditions of the equations! To know which solution you want, you have to feed in an x value. This makes the y value dependent on the x value. Attempting to solve a dependent system of linear equations via numpy will throw an error, because the solution depends on your inputs.

Independence means that no equation in the system can be algebraically derived from the others. In systems of linear equations with 2 equations, this means that the lines only intersect at one point, or the equations reflect parallel lines. There is either no solution or a single solution.

Supplemental Reading hereLinks to an external site.

Consider the following sets of systems of equations in the code cells

Your task for this question is as follows:

Determine which system of equations above is linearly independent. Indicate which system is linearly independent by assigning the string "system 1" or "system 2" to the variable ans8a. For the system that is linearly independent, solve the system of equations and assign the solution to the variable ans8b. Round all values to 3 decimal places. Note: There are no restrictions on using functions from Python libraries for this question.

#systemofequations1

A1=np.array([

[7,0,0],

[4,6,3],

[0,4,2]])

b1=np.array([4,8,5])