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MAST10007 Linear Algebra, Semester 1 2025
Written assignment 4
Student Number
Submit your assignment solutions together with this coversheet via the MAST10007 Gradescope website before 12pm noon AET on Thursday, 1st May.
• No extensions will be granted!
• You need to apply for an exemptions in the STOP1 Special Consideration Portal!
• Late submissions are only possible up to 10 hours after the deadline with a deduction of 1 mark (5% of the total marks of this assignment) per hour after the deadline!
• This assignment is worth 3% of your final MAST10007 mark.
• Note, that we take only the 5 best written assignments of the 6 you need to write into account for the final mark.
• Assignments must be neatly handwritten unless you have a medical exception. This, however, includes digitally handwritten documents using an ipad or a tablet and stylus, which have then been saved as a pdf.
• You must use methods taught in MAST10007 Linear Algebra to solve the assignment questions.
• Full working must be shown in your solutions.
• Marks will be deducted in every question for incomplete working, insufficient justification of steps and incorrect mathematical notation.
• Part of your overall mark is for quality of exposition meaning full sentences, explanation of your computations and explicitly mentioning what your result is. Have a look at the solutions of the tutorial sheets to get an idea of how detailed we want to have it!
• There are in total 20 points to achieve in this assignment.
• Begin your answer for each question on a new page!
Please, turn the page for the other questions!
1. Summary 4 points.
Summary three lectures chosen from Lecture 14 to Lecture 18. Each summary should not be longer than 3 full sentences and should not involve formulas. Note, that any object or technique introduced is worth mentioning. Use the space below. Clearly indicate which lecture you are writing about.
2. Inverse of matrices – 8 points. Consider the 4 × 4 matrix A given by
(a) Compute A−1 using the reduced row-echelon form.
(b) Using the previous question, solve the system (S) of linear equations defined below
Hint: relate first the matrix involved in this equation to the matrix A from part (a). You can use the properties of the inverse that are on the tutorial sheet of week 7.
3. Trigonometric functions – 8 points.
(a) Show that the set of functions on R defined below is linearly independent.
B = {g1(x) = cos(x), g2(x) = sin(x), g3(x) = cos(2x), g4(x) = sin(2x)}.
Hint: have a look at various values of x ∈ R to check whether you can find common coefficients to show the linear independence. You do not necessarily need to use augmented matrices if you choose x in a smart way.
(b) Consider the basis C of span{B}, defined by
C = {f1(x) = cos(x) + cos(2x),
f2(x) = cos(x) − sin(x) + 2 sin(2x),
f3(x) = 2 cos(x) + cos(2x),
f4(x) = − sin(x) + 2 cos(2x) + sin(2x)}.
Compute the coordinate transformation matrices PB,C and PC,B for the coordinate changes from C to B and its converse. Moreover, verify that 