Math 2568/4568 Midterm
(Summer 2021)
Testing guidelines
•) Show all of your work. Correct answers with no supporting work may receive no credit.
•) Circle or highlight your final answers.
•) You may use your textbook, notes, posted resources, and MATLAB/Octave.
Problem Maximum Recorded
Number Point Value Point Value
Problem 1 10
Problem 2 13
Problem 3 12
Problem 4 11
Problem 5 12
Problem 6 12
Problem 7 10
Problem 8 10
Problem 9 10
Total 100
Problem 1 Three vectors v1,v2,v3 in R3 are given by
v1 =
13
2
, v2 =
1a
1
, v3 =
27
b
i) [4 points] Form the matrix A = [v1 v2 v3] and put it into upper triangular form using only type I and
III row operations that exist for all values of a and b.
ii) [3 points] Using part i), find an equation relating a and b which indicates exactly when the set of three
vectors is linearly dependent.
iii) [3 points] If a = 3 and b = 5, determine if the set of vectors {v1,v2,v3} is a basis for R3. You should
explain your reasoning.
Problem 2 Let L : P4 → P4 be the map defined by L(p) = L(p(x)) = x2p′′(x)− 2p(x)− 5p′(x) (where
P4 denotes the vector space of polynomials with real coefficients of degree less than 4).
i) [4 points] Find a matrix A satisfying SL(p) = A ∗ Sp, where S = {1, x, x2, x3} is the standard basis
for P4.
ii) [3 points] Using i), explain why this implies L is a linear transformation.
iii) [3 points] Find a basis for ker(L).
iv) [3 points] Find a basis for im(L).
Problem 3 A system of equations is given by
3×1 − x2 + 5×3 + 2×4 = 12
2×1 − 3×2 + 8×3 − 7×4 = 10
−4×1 + 10×2 − 4×3 + 5×4 = 1
i) [4 points] Write down the augmented coefficient matrix A. Then using row operations, put the ACM
into reduced row echelon form using rational number format. When doing this, you should indicate the
exact row operation you are using at each stage, and the matrix it produces (in other words, you should
write down the sequence of matrices that are produced at each step of the row reduction).
ii) [4 points] Write down the set of equations corresponding to rref(A) and determine the set of solutions.
If there are infinitely many solutions, express your answer in parametrized form.
iii) [4 points] Using ii), write down a description of the solution set as {xh + xp} where xp is a particular
solution, and xh is the general solution to the assoicated homogeneous equation. You should describe
explicitly what xp and xh are.
Problem 4 Let L : R3 → R4 be the linear transformation given by
L
x1x2
x3
=
(3×1 − 2×2 − 7×3)
(5×1 − 3×3)
(4×2 − 3×3)
(6×1 + 2×2 − 3×3)
i) [4 points] Find a matrix A such that L(x) = A ∗ x for all x ∈ R3.What is the relation between A and
the matrix representation eLe of L with respect to the standard bases for R3 and R4?
ii) [3 points] Let S = {v1,v2,v3} be the basis for R3 where
ev1 =
10
1
, ev2 =
01
2
, ev3 =
13
0
Write down the base transition matrix eTS . How is eTS related to STe?
iii) [4 points] Compute the matrix representative eLS of L.
Problem 5 Let A =
[
7 2
−5 2
]
.
i) [4 points] Apply row operations to [A | I] to find the inverse of A.
ii) [4 points] Use your work to express A−1 explicitly as a product of elementary matrices. Use the correct
notation for each elementary matrix, as it is given in the textbook.
iii) [4 points] Using ii), express the original matrix A explicitly as a product of elementary matrices (again
using correct notation).
Problem 6 Let A =
3 −2 5 7 −10 9 −4 6 10
−2 1 9 −7 6
.
i) [4 points] Find a basis for the nullspace N(A) of A.
ii) [4 points] Find a basis for the column space C(A).
iii) [4 points] Determine the rank and nullity of A.
The following statements are either true or false. Indicate which, and justify your answer.
Problem 7 [10 points] If A and B are two matrices in R2×2 where A is invertible and B has four
completely different entries with each entry an invertible number, then A ∗B is also invertible.
Problem 8 [10 points] A set consisting of three vectors {v1,v2,v3} is linearly independent precisely
when each pair of vectors in the set is linearly independent.
Problem 9 [10 points] If A and B are two real matrices where A ∗ B is non-singular, then both A and
B are non-singular.