PRACTICE MAT223H1S Final Exam
Winter 2025
Section A.
Instructions:
1. The problems in this section will ask you to complete a definition or to prove a theorem from the course lecture notes.
2. Definitions must be stated precisely as they are in the course lecture notes (up to rewording). Each definition statement is worth one point and no partial credit will be given.
3. Theorem proofs will each be worth five points, which will be awarded using our standard rubric (which is available in the Section C instructions).
A1. (1 point) Complete the following definition: The span of vectors , , . . . , in Rn is . . .
A2. (1 point) Complete the following definition: A system of linear equations is called consistent if . . .
A3. (1 point) Complete the following definition: The dimension of a vector space is . . .
A4. (1 point) Complete the following definition: A non-zero vector is called an eigenvector of a matrix A if . . .
A5. (1 point) Complete the following definition: The singular values of a matrix are . . .
A7. (5 points) Prove the following Proposition (Proposition 11.10):
Let B be an orthonormal basis for R
n and take any vectors , in R
n. Then, []B · []B = · .
Section B.
Instructions:
1. Each problem in this section is worth three points.
2. Problems with multiple parts will be worth one point each. Otherwise, no partial credit will be given.
3. You do not need to show your work or provide justification on any problem in Section B.
4. Your answer must be placed in the answer box provided.
5. We have provided extra space for your scratch work on each problem, but nothing outside of the answer box will be considered toward your score on the Section B problems.
B1. (3 points) For the systems of linear equations described below, determine whether the system has no solution, exactly one solution, or infinitely many solutions.
a) A system whose coefficient matrix is invertible.
Answer: The system of linear equations has
No solutions
Exactly one solution
Infinitely many solutions
b) A system whose augmented matrix is invertible.
Answer: The system of linear equations has
No solutions
Exactly one solution
Infinitely many solutions
c) The system with augmented matrix AT
, where A is the augmented matrix representing the system in part (a).
Answer: The system of linear equations must have
No solutions
Exactly one solution
Infinitely many solutions
B2. (3 points) For each linear transformation defined below, determine whether the reduced row echelon form. of its defining matrix has a pivot in every row, every column, both, or neither.
a) F : R4 → R3
satisfying that is linearly independent.
Answer:
Pivots in every row and every column
Pivots in every row but not every column
Pivots in every column but not every row
None of the above
b) G = TQ, where Q is an n × n orthogonal matrix.
Answer:
Pivots in every row and every column
Pivots in every row but not every column
Pivots in every column but not every row
None of the above
c) H : R2 → R4 with
Answer:
Pivots in every row and every column
Pivots in every row but not every column
Pivots in every column but not every row
None of the above
B3. (3 points) Calculate the following determinants.
a) det(AB) where and B = AT
, the transpose of A.
det(A) =
b) det(F), where F is the inverse of the function G where G: R2 → R2
is the linear transformation which stretches vectors in the + direction by -3 and leaves the − direction unchanged.
det(F) =
c) Let C be standard defining matrix of F from part (b). Is it possible that C similar to the matrix AB from part (a)?
Yes, it is possible
No, it is not possible
B4. (3 points) Determine which of the following matrices are invertible. If there is not enough information to determine whether the matrix is invertible or not invertible, select “could be either”.
a) A 3 × 3 matrix N satisfying that N3
is the zero matrix.
Is invertible
Is not invertible
Could be either
b) A symmetric matrix.
Is invertible
Is not invertible
Could be either
c) The defining matrix of the linear transformation in R
3
that rotates around the z-axis by an angle of .
Is invertible
Is not invertible
Could be either
B5. (3 points) Chapter 10, part B problem (Kevin)
Let A be a 3 × 3 matrix with eigenvalues 0, 1, 2. Find the eigenvalues of the following matrices:
a) The matrix A2.
b) The matrix A − I3.
c) The matrix 3(AT
)2.
B6. (3 points) Let ε be the standard basis of R3. Consider the basis
a) Find given that
b) Find the change of basis matrix from B to ε.
c) Find the change of basis matrix from ε to B.
B7. (3 points) Let
a) Find the characteristic polynomial of A.
Answer: χA(x) =
b) Find the dimension of the 1-eigenspace E1.
Answer: dim E1 =
c) Find an invertible matrix C so that C−1AC is a diagonal matrix.
B8. (3 points) Determine which of the following statements are always true and which are always false. If there’s not enough information to determine whether a statement is always true or always false, select “could be true or false”.
a) If and are two vectors in Rn such that · = 0, then {, } is a linearly independent set.
Always true
Always false
Could be true or false
b) If a linear transformation F : R3 → R3 preserves the angle between every pair of vectors, then its defining matrix A is orthogonal.
Always true
Always false
Could be true or false
c) Let V be a vector subspace of Rn. Every basis of V has n elements.
Always true
Always false
Could be true or false
Section C.
Instructions:
1. Each problem in this section is worth 5 points.
2. You must provide justification for all of your answers in Section C.
3. Points will be awarded based on the rubric below. Note that half points may be awarded, and further rubric items may be added to cover potential cases not outlined below.
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Points
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Rubric
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5
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Solution is presented with clear justification that is logically complete and correct. May include minor typos and computational errors if they do not majorly impact the argument. No important steps are missing or assumed. All assumptions and special cases have been covered. All suggestions for improvement come under the category of “improvements for clarity” rather than “correcting logical errors”. Omission of details will be judged depending on context of the material, with simpler steps being acceptable for omission when covering more advanced topics.
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4
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Solution is close to full and complete, but contains either a computational error or an error in reasoning that majorly impacts the argument. This score is also appropriate for solutions that are mathematically sound but confusingly written.
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3
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Solution is incorrect, but understanding of the problem was demonstrated and stu- dent provided a clear outline of a potential approach with information about where they got stuck -or- solution is correct but justification is insufficient or so confus- ingly written that it cannot be followed with a reasonable amount of effort.
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2
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Solution is incorrect, but student demonstrated understanding of the problem -or- solution is correct and student did not provide justification for their answer.
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1
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Solution is incorrect and student did not demonstrate understanding of the problem, but did demonstrate some knowledge of relevant material.
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0
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Solution is incorrect or incomplete, and there was no demonstration of knowledge of relevant material.
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C1. (5 points) Let A be an m × (n + 1) matrix, and suppose that the system of linear equations in n variables with augmented matrix A has at least one solution. Show that the homogeneous system of linear equations in n + 1 variables with coefficient matrix A has infinitely many solutions
C2. (5 points) Let B = {, , . . . , } be a set of vectors in R
4
. Then B cannot be a basis of R4
. If true, provide a proof. If false, provide a counterexample, and justify why this is one.
C3. (5 points) Let F : Rn → Rn be a linear transformation and B a basis for R
n. Show that if the defining matrix AF is invertible, then AF,B is also invertible.
C4. (5 points) Let A be an 6×7 matrix. Is it possible that the nullity of A equals the nullity of its transpose AT
? If yes, find an example and prove that it is an example. If no, prove it.