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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu Student number Semester 2 Assessment, 2021
School of Mathematics and Statistics MAST10007 Linear Algebra
Reading time: 30 minutes — Writing time: 3 hours — Upload time: 30 minutes
This exam consists of 25 pages (including this page) with 12 questions and 125 total marks Permitted Materials
•This exam and/or an offline electronic PDF reader, one or more copies of the masked
exam template made available earlier and blank loose-leaf paper.
•One double sided A4 page of notes (handwritten only).
•No calculators are permitted. No headphones or earphones are permitted. Instructions to Students
•Wave your hand right in front of your webcam if you wish to communicate with the
supervisor at any time (before, during or after the exam).
•You must not be out of webcam view at any time without supervisor permission.
•You must not write your answers on an iPad or other electronic device.
•Off-line PDF readers (i) must have the screen visible in Zoom; (ii) must only be used to
read exam questions (do not access other software or files); (iii) must be set in flight mode
or have both internet and Bluetooth disabled as soon as the exam paper is downloaded. Writing •Marks are awarded for
–Using appropriate mathematical techniques.
–Showing full working, including results used. –Accuracy of the solution.
–Using correct mathematical notation.
•If you are writing answers on the exam or masked exam and need more space, use blank
paper. Note this in the answer box, so the marker knows.
•If you are only writing on blank A4 paper, the first page must contain only your student
number, subject code and subject name. Write on one side of each sheet only. Start each
question on a new page and include the question number at the top of each page. Scanning and Submitting
•You must not leave Zoom supervision to scan your exam. Put the pages in number
order and the correct way up. Add any extra pages to the end. Use a scanning app to
scan all pages to PDF. Scan directly from above. Crop pages to A4. Page •Su 1bmio t fyou2 r s5can — ned ex aad m ad s a a sinngly e P e DF xfitleraa nd p car a ef g ull e y rsevi a ew ftthe e rsub p misasig on ein
Gradescope. Scan again and resubmit if necessary. Do not leave Zoom supervision until 25 — P
you have confirmed orally with the supervisor that you have received the Gradescope confirmation email.
•You must not submit or resubmit after having left Zoom supervision. c
University of Melbourne 2021 Page 1 of 25 pages
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question1(11marks) Consider the linear system x+ 2y+ 3z= 6 4x+ 5y+ 6z= 15 7x+ 8y+ 9z= 24
(a) Reduce the augmented matrix [A|b] of the system to reduced row-echelon form. Indicate
the row operations used in each step.
(b) Determine the ranks of Aand [A|b] and interpret your answer.
(c) Find the solution set of the system.
(d) Find the solution set of the corresponding homogeneous system.
(e) Prove that the set of all choices for csuch that [A|c] can be solved, forms a plane through the origin in R3.
(f) Find a normal vector to the plane in part (e).
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question2(8marks)
(a) Let Abe an n×nmatrix. Prove that the determinant of Aequals zero if and only if the
nullity of Ais greater than or equal to 1.
(b) Let u∈R3, and consider the linear transformation T:R3→R3such that T(v) = u×v.
What is the determinant of [T]? Justify your answer.
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question3(14marks) Let L1be the line x−3 −1=y−8−2=z+ 2 and let L2be the line
x= 4 −3s, y =−1+5s, z =−4+4s, s ∈R.
(a) Find the point of intersection of L1and L2.
(b) Find the angle between the lines L1and L2.
(c) Find a Cartesian equation of the plane in R3that contains both L1and L2.
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question4(9marks)
Let V=R2be the real vector space with vector addition ⊕defined by
(x1, x2)⊕(y1, y2)=(x1+y1−1, x2+y2−2) for each (x1, x2),(y1, y2)∈V
and scalar multiplication defined by
α(x1, x2)=(αx1−α+ 1, αx2−2α+ 2)
for each (x1, x2)∈Vand α∈R.
(a) Verify the associative property u⊕(v⊕w) = (u⊕v)⊕wwhen u= (−3,1),v= (5,4),w= (2,7).
(b) Prove 1 u=ufor all u∈V.
(c) Prove the distributive property (α+β)u= (αu)⊕(βu) for all u∈Vand α, β ∈R.
(d) Find the zero vector 0in V. Show that u⊕0=ufor all u∈V.
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question5(12marks)
Determine if the following sets are subspaces of the real vector space M2 2of 2 × , 2 matrices.
If the set is a subspace, you must use the subspace theorem to prove it. If the set is not a
subspace, then you must provide an explicit counter-example.
(a) The set Sof all matrices Asuch that AD=DA, where D= 1 2 3 4 .  
(b) The set Mof all matrices of the form s+ 3 0 s 2 + t t + 6 where s, t ∈R.  
(c) The set Wof all matrices of the form r0 0r2 where r∈R.  
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question6(12marks)
Let Wbe the subspace of P3spanned by
S={1+2x+ 3x3,2+4x+ 6x3,1 + x2+x3,5+3x+ 5x2+ 8x3,3−5x+x2−2x3}. You may assume that A= 1 2 1 5 3  ∼ 1 2 0 0 2    2 4 0 3 −5 0 0 1 5 1      =B. 0 0 1 5 1   0 0 0 3 −9      3 6 1 8 −2 0 0 0 0 0 (a) Is Wequal to P3? Explain.
(b) Find a basis Bfor Wconsisting of polynomials in S.
(c) What is the dimension of W?
(d) For each vector in Sthat is not in B, find the coordinate vector with respect to the basis B.
(e) Let C={2+4x+ 6x3,1 + x2+x3,3−5x+x2−2x3}.
(i) Is Clinearly independent? Explain.
(ii) Is Ca basis for W? Explain.
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question7(10marks)
Let P2be the vector space of real polynomials of degree ≤2 with ordered basis B={1, x, x2}.
Let T:P2→R4be the linear transformation defined by
T(p(x)) = (p(0),p(1),p(−1),p(3)). (a) What is T(x2−3x)?
(b) Find the matrix representation of Twith respect to the basis Band the standard basis S of R4. (c) Compute the rank of T.
(d) Is Tsurjective? Explain your answer.
(e) Is Tinjective? Explain your answer.
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question8(9marks)
For each of the following linear transformations, determine the (i) image (ii) kernel
(iii) inverse transformation, if it exists.
(a) Reflection R:R2→R2in a line `⊂R2passing through the origin.
(b) The map P:R3→R3given by orthogonal projection onto a plane Π through the origin in R3.
(c) The differentiation function D:P3(R)→ P2(R) mapping a polynomial p(x) to its deriva- tive dp dx .
Hint: You do not need to find the matrix representations of the transformations.
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question9(14marks) Consider the complex matrix A= 6 2− 2i+ 4 2 . i  
(a) Show that Ais a Hermitian matrix. (b) Find the eigenvalues of A.
(c) Find the eigenspace corresponding to each eigenvalue of A.
(d) Find a unitary matrix Uand a diagonal matrix Dsuch that A=U DU 1 − .
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021
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MAST10007 Final Exam S2 2021 - Linear Algebra Assessment Details - Studocu MAST10007 Linear Algebra Semester 2, 2021 Question10(8marks)
(a) Let Abe an arbitrary 2 ×2 matrix.
(i) Determine the characteristic equation of A.
(ii) Show that the characteristic equation of Acan be written as λ2−Tr(A)λ+ det(A)=0.
(b) Let Vbe a real vector space equipped with an inner product hu,vi. Using the inner
product space axioms, prove that for any u,v∈V, ku+vk2+ku−vk2= 2kuk2+ 2kvk2.
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