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Final Exam Practice Problems
MTH 301-summer 2022
Q1)True or False:
1) The set ℝ is a vector space (
)
2) ℝ is a subspace of ℝ
(
)
3) The set ℚ, are filed
4) The additive identity in ℝ is (0,0,0)
(
)
5) For the set S = {(2,-1,0),(5,3,4),(-1,1,2)}, we can write one vector as a linear combination of the
others as follow (2, −1,0) = 3(5,3,4) − (−1, 1, 2)
(
)
6) The additive identity in ℝ is (0,0,0)
(
)
7) S={( , , 0) : , ∈ ℝ} is a subspace of ℝ . (
)
Q2) Determine whether the following sets of vectors in ℝ are linearly independent or linearly
dependent
a) S = {(1, 2),(2, 4)},
b) S = {(1, 0),(0, 1),(−2, 5)}.
Q3) Let S={(1,2,3),(0,1,2),(-2,0,1)}. Is S linearly independent or linear dependent set in ℝ ?
Q4) find a basis and a dimension of the following sets of ℝ :
a) W={(s+4t, t, s, 2s-t): s and t are real numbers}.
b) W={ ( 0, 6t , t , -t) : t is a real number }
−2 −4 4
5
Q5) = 3
6 −6 −4 Find the followings:
−2 −4 4
9
a) The rank
b) The row space
c) The column space
Q6) =
1
0
0
Find the followings of A :
2
(a) The rank
(b) The row space
(c) The column space.
Final Exam Practice Problems
MTH 301-summer 2022
Q7) Fill in the blank
a) The length of a vector is also called its ……….
b) If ||v|| = 1, then the vector v is called …………
c) The dot product of two vectors is…………….
d) Cauchy-schwarz Inequality for u and v……………..
e) The vector………………. is orthogonal to every vector.
f) Two vectors u and v in Rn are orthogonal if ………………
g) If A and B are ………….. matrices, then they have the same eigenvalues.
Q8) (a) Find the length of the following vectors :
(1) v=(1,2,2)
(2) v=(2,0,-5,5)
(b) Find the distance between and Find the dot product of and
(1) =(2,-1,0) and =(1,4,1)
(2) =(0,1,2,3) and =(1,0,4,-1)
Q9) Find the dimensions of ℝ and ℝ for the linear transformation represented by each
matrix: The linear transformation T : ℝ → ℝ is defined by T(v) = Av.
a) = −
b) =
Q10) Let =
−
−
Find the characteristic equations .
Find the eigenvalues of A
Find the eigenvectors of A
Determine whether the matrix A is diagonalizable or not .
Final Exam Practice Problems
MTH 301-summer 2022
Q11) Let =
−
Find the characteristic equations of A .
Find the eigenvalues of A
Find the eigenvectors of A
Determine whether the matrix A diagonalizable or not
Find an invertible matrix such that
=
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