Chapter 4 exercises: Linear Algebra and Matrix Operations
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Attempt these tasks before opening the worked answers. Use the chapter experiment notebook, or import NumPy as np and Pandas as pd in your own notebook. Retain your first attempt and explain each correction.
B2-04-E01 · Check a linear solution
Solve Ax=b and verify the residual with a numerical tolerance.
A=np.array([[2.,1.],[1.,3.]])
b=np.array([8.,9.])One hint after an attempt
Verify by substitution.
B2-04-E02 · Recognize dependent columns
Find the matrix rank and explain why two independent coefficient effects cannot be identified from these columns alone.
X=np.array([[1.,2.],[2.,4.],[3.,6.]])One hint after an attempt
Express one column as a multiple of another.
B2-04-E03 · Use least squares
Fit an intercept and slope using lstsq, then check the fitted values.
x=np.array([1.,2.,3.])
y=np.array([3.,5.,7.])One hint after an attempt
Build a column of ones for the intercept.
Reflect and get feedback
Explain one result without looking at the code. Identify one boundary case that would change your method. An optional tutor prompt is: “Here is my attempt at B2-04-E03. Give one hint about the violated assumption; do not give the solution yet.” Then use the worked answers.