Midterm 2

Question 1

Find the length of u = (-1,1,3,-5)

Answer 1

sqrt(1^2 + 1^2 + 3^2 + 5^2) = 6

Question 2

What are the two other ways to say “Find the length of a vector u”?

Answer 2

1) Find the Euclidean norm/length of u
2) Find ||u||

Question 3

Find a unit vector in the direction u = (1, -5, 3, 2)

Answer 3

= u/||u||
= (1,-5,3,2)/sqrt(1^2 + 5^2 + 3^2 + 2^2)

Question 4

Find the Euclidean distance between u = (-1,0,3) and v = (2,1,2) or find d(u, v)

Answer 4

= ||v - u||
= sqrt((2 - (-1))^2 + (1 - 0)^2 + (2 - 3)^2)
= sqrt(11)

Question 5

Given u = (2, 0, -1) and v = (4, 0, 1), which of the following are true?

a) u and v are parallel
b) u is a unit vector
c) ||v|| = 5
d) u * v = 7

Answer 5

a) False, they’re not multiples of each other
b) False, the length of u is not 1
c) False, ||v|| = sqrt(17)
d) True, u * v = 8 - 1 = 8

Question 6

Find the angle between vectors u = (5, 3) and v = (4, -1).

Answer 6

Use u * v = ||u|| ||v|| cos(theta) and rearrange for theta

Question 7

When are two vectors orthogonal and what does this mean?

Answer 7

Two vectors are orthogonal or perpendicular if the dot product is zero

Question 8

Are any of the given vectors orthogonal to one another?
u = (2,-1,2,1)
v = (1,1,-1,1)
w = (1,1,3,1)

Answer 8

u * v = 0 (yes)
u * w = 8 (no)
v * w = 0 (yes)

Question 9

Given u = (2,-1,3) and a = (1,0,2) find,
a) the vector component of u along a
b) the vector component of u orthogonal to a

Answer 9

a) (1,0,2)
b) (2/5, -1, -1/5)

Question 10

What is the point-normal form of a equation of the
plane? explain how to get the 3 values.

Answer 10

a(x - x0) + b(y - y0) + c(z - z0) = 0
(a, b, c) = normal vector of the plane
(x, y, z) = arbitrary point on the plane
(x0, y0, z0) = a known point on the plane

Question 11

Find the equation of the plane that’s parallel to the plane 4x - 2y + 5z = 6 that goes through the point (-1, 0, 8).

Answer 11

4(x - (-1)) - 2(y - 0) + 5(z - 8) = 0

Question 12

What is the equation of the plane in standard form?

Answer 12

ax + by + cz = d

Question 13

What is the parametric equations of a line L given (x, y, z) = (x0, y0, z0) + t(a, b, c)

Answer 13

x = x0 + ta
y = y0 + tb
z = y0 + tc

Question 14

Find the parametric equations of the line L containing the point p = (1, -2, 3) and perpendicular to the plane 2x - y + 3z = 5

Answer 14

If L is perpendicular to the equation then it is parallel to the normal vector n = (2, -1, 3)
x = 1 + 2t
y = -2 - t
z = 3 + 3t

Question 15

Which of the following statements are true?

a) The planes 3x + y - 2z = 7 and 6(x - 2) + 2y - 4(z + 1) = 0 are parallel

b) The planes 7x - y + 3z = 6 and 2x + 7y + 4z = 0 are perpendicular

c) The lines x = 3 + 2t, y = 5 - t, z = 6 + 3t and x = 7 - 2t, y = 9 + t, z = 1 - 3t are parallel

d) The line x = 3 + 2t, y = 5 - t, z = 6 + 3t is perpendicular to the plane 2x - y 3z = 5

Answer 15

a) True
b) False
c) True
d) False

Question 16

Find the cross product of (6,3,7) x (2,9,0)

Answer 16

= (-63, 14, 48)

Question 17

Find the area of the parallelogram determined by the vectors u = (2, -1, 3) and v = (1, 5, 1)

Answer 17

u x v = sqrt(378)

Question 17

What makes up a vector space?

Answer 17

1) The objects {(u1, u2, u3, …, un)}
2) Closed under addition u + v
3) Closed under scalar multiplication ku

Question 17

For which of the following sets will vector space axiom 6 hold? ku belongs to V.

a) Vectors in the form (a, b) where a and b are positive real numbers

b) Vectors of the form (a, 3, b)

c) Upper triangular matrices

d) 2 x 2 matrices with a determinant of 0

Answer 17

a) No
b) No
c) Yes
d) Yes

Question 18

For which of the following sets will vector space axiom 5 hold? u + (-u) = (-u) + u = 0

a) Vectors in the form (a, b, 3)
b) Vectors of the form (a, b, 0)
c) Diagonal matrices
d) 2x2 matrices where all the entries are positive
e) symmetric matricies

Answer 18

a) No
b) Yes
c) Yes
d) No
e) Yes

Question 19

Let V be the set of all points on the line y = x + 1. Does this form a vector space?

Answer 19

No,
v = (5, 6)
u = (1, 2)

u + v = (6, 8) not in V

Question 20

For which of the following sets will vector space property 1 hold? u + v is in V

a) (a, b, 1)
b) (a, b, 0)
c) Diagonal matrices
d) 2x2 matrices, all positive

Answer 20

a) No
b) Yes
c) Yes
d) Yes

Question 21

When is w a sub space

Answer 21

a) closed under addition u + v is in W

b) closed under scalar multiplication ku is in W

Question 22

Which of the following are subspaces of the indicated vector space?

a) Is the set of all 2x2 matrices A of the form|-1 b| a subspace of M22?
| 0 c|

b) (a, b, 0) with b = 3a a subspace of R^3

c) {a0 + a1x + a2x^2 | a0 + a1 = 0} a subspace of P2

Answer 22

a) No
b) Yes
c) Yes

Question 23

Consider u = (1,-1,1) & v = (-1,2,1), show that…

a) w = (1,0,3) is a liner combination of u and v

b) x = (1, 0, 0) is NOT a linear combination of u and v

Answer 23

a)
w = k1 u + k2 v
turn into matrix and get k1 = 2, k2 = 1

b) get R3 = [0 0 | -3] which means no solution

Question 24

Which of the following statements are true?

a) Five vectors must span R^3
b) It is possible 5 vectors span R^4
c) It is possible 5 vectors span R^7

Answer 24

a) False
b) True
c) False

Question 25

Determine weather the following vectors span R^3
v1 = (2, 1, -3)
v2 = (0, -5, 1)
v3 = (6, -7, -9)

Answer 25

(a, b, c) = k1 v1 + k2 v2 + k3 v3

2 k1 + 6 k3 = a
k1 - 5 k2 - 7 k3 = b
-3 k1 + k2 - 9 k3 = c

det(A) = 20

The system has a unique solution since det does not equal zero so it spans R^3

Question 26

What is a linearly independent set of vectors?

Answer 26

If no vector in the set can be expressed as a linear combination of the others.

Question 27

What is a linearly dependent set of vectors?

Answer 27

If there is a vector in the set can be expressed as a linear combination of at least one of the others.

Question 28

Are these vectors linearly dependent or independent?
v1 = (1,0,-1)
v2 = (5,2,8)
v3 = (6,2,7)

Answer 28

v3 = v1 + v2

they are linearly dependent.

Question 29

How do we test if a given set of vectors is linearly independent

Answer 29

1. write the linear combination of the given vectors equal to zero
2. Set up the corresponding linear system
3. Check there is only the trivial solution in the system

a) Row reduce to find the trivial solution

b) if square matrix find det not equal to 0

Question 30

Are these P2 vectors linearly in/dependent?
p1 = 1 - x + x^2
p2 = 1 - x^2
p3 = 1 + x

Answer 30

k1 = 0, k2 = 0, k3 = 0
L. I.

Question 31

For the following questions below answer if they are linearly dependent/independent

a) a finite set of vectors that contains the zero vector?

b) A set with any vector that is not the zero vector?

c) a set with exactly two vectors that are not scalar multiples of each other

Answer 31

a) LD
b) LI
c) LI

Question 32

Which of the following sets are linearly independent?

(a) {(3, 1, 0, 2), (2, -2/3, 0, 4/3 )}
(b) {(2, 1, 2), (1, 2, 3)}
(c) {(5,3,0)}
(d) {6 + 2x - 2x^2, 3 + x - x^2}

Answer 32

a) LD
b) LI
c) LI
d) LD

Question 33

Example: Which of the following statements are true?

(a) Suppose you have 5 vectors in R^4. They must be linearly dependent.

(b) Suppose you have 8 vectors in R^9. They must be linearly independent.

(c) Suppose you have 6 vectors in R^3. They could be linearly independent

Answer 33

a) True
b) False
c) False

Question 34

What are the two conditions for a basis of the set S
S = {v1, v2, v3}

Answer 34

1) S is linearly independent
2) S spans V

Question 35

Determine whether the vectors (1,-3,2), (-4,1,0), and (-1,2,1) from a basis for R^3

Answer 35

1) Spanning
det = -25 so has solutions for k1, k2, and k3

2) The system is LI because the system Ax = 0 has only the trivial solution.

Question 36

Find the coordinate vector for
m = |-1 3|
|2 0|

Answer 36

(-1,3,2,0)

Question 37

Given, the following:
p1 = 1- x^2
p2 = 2 + x
p3 = -1 + 2x + x^2

find:
a) The polynomial p whose coordinate vector with respect to the basis S is (p)s = (-1,2,-1)

b) The coordinate vector of p =1 + 2x - 2x^2

Answer 37

a) p = -1 p1 + 2 p2 - 1 p3 = 4
b) 1 + 2x - 2x^2 = k1 p1 + k2 p2 + k3 p3, k1 = 13/4, k2 = -1/2, and k3 = 5/4

Question 38

a) If a set has more than n vectors, then?

b) If a set has fewer than n vectors, then?

Answer 38

a) it’s linearly dependent
b) does not span V