Midterm 2 Flashcards
Find the length of u = (-1,1,3,-5)
sqrt(1^2 + 1^2 + 3^2 + 5^2) = 6
What are the two other ways to say “Find the length of a vector u”?
1) Find the Euclidean norm/length of u
2) Find ||u||
Find a unit vector in the direction u = (1, -5, 3, 2)
= u/||u||
= (1,-5,3,2)/sqrt(1^2 + 5^2 + 3^2 + 2^2)
Find the Euclidean distance between u = (-1,0,3) and v = (2,1,2) or find d(u, v)
= ||v - u||
= sqrt((2 - (-1))^2 + (1 - 0)^2 + (2 - 3)^2)
= sqrt(11)
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
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
Find the angle between vectors u = (5, 3) and v = (4, -1).
Use u * v = ||u|| ||v|| cos(theta) and rearrange for theta
When are two vectors orthogonal and what does this mean?
Two vectors are orthogonal or perpendicular if the dot product is zero
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)
u * v = 0 (yes)
u * w = 8 (no)
v * w = 0 (yes)
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
a) (1,0,2)
b) (2/5, -1, -1/5)
What is the point-normal form of a equation of the
plane? explain how to get the 3 values.
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
Find the equation of the plane that’s parallel to the plane 4x - 2y + 5z = 6 that goes through the point (-1, 0, 8).
4(x - (-1)) - 2(y - 0) + 5(z - 8) = 0
What is the equation of the plane in standard form?
ax + by + cz = d
What is the parametric equations of a line L given (x, y, z) = (x0, y0, z0) + t(a, b, c)
x = x0 + ta
y = y0 + tb
z = y0 + tc
Find the parametric equations of the line L containing the point p = (1, -2, 3) and perpendicular to the plane 2x - y + 3z = 5
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
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
a) True
b) False
c) True
d) False
Find the cross product of (6,3,7) x (2,9,0)
= (-63, 14, 48)
Find the area of the parallelogram determined by the vectors u = (2, -1, 3) and v = (1, 5, 1)
u x v = sqrt(378)
What makes up a vector space?
1) The objects {(u1, u2, u3, …, un)}
2) Closed under addition u + v
3) Closed under scalar multiplication ku
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
a) No
b) No
c) Yes
d) Yes
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
a) No
b) Yes
c) Yes
d) No
e) Yes
Let V be the set of all points on the line y = x + 1. Does this form a vector space?
No,
v = (5, 6)
u = (1, 2)
u + v = (6, 8) not in V
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
a) No
b) Yes
c) Yes
d) Yes
When is w a sub space
a) closed under addition u + v is in W
b) closed under scalar multiplication ku is in W
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
a) No
b) Yes
c) Yes
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
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
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
a) False
b) True
c) False
Determine weather the following vectors span R^3
v1 = (2, 1, -3)
v2 = (0, -5, 1)
v3 = (6, -7, -9)
(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
What is a linearly independent set of vectors?
If no vector in the set can be expressed as a linear combination of the others.
What is a linearly dependent set of vectors?
If there is a vector in the set can be expressed as a linear combination of at least one of the others.
Are these vectors linearly dependent or independent?
v1 = (1,0,-1)
v2 = (5,2,8)
v3 = (6,2,7)
v3 = v1 + v2
they are linearly dependent.
How do we test if a given set of vectors is linearly independent
- write the linear combination of the given vectors equal to zero
- Set up the corresponding linear system
- 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
Are these P2 vectors linearly in/dependent?
p1 = 1 - x + x^2
p2 = 1 - x^2
p3 = 1 + x
k1 = 0, k2 = 0, k3 = 0
L. I.
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
a) LD
b) LI
c) LI
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}
a) LD
b) LI
c) LI
d) LD
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
a) True
b) False
c) False
What are the two conditions for a basis of the set S
S = {v1, v2, v3}
1) S is linearly independent
2) S spans V
Determine whether the vectors (1,-3,2), (-4,1,0), and (-1,2,1) from a basis for R^3
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.
Find the coordinate vector for
m = |-1 3|
|2 0|
(-1,3,2,0)
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
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
a) If a set has more than n vectors, then?
b) If a set has fewer than n vectors, then?
a) it’s linearly dependent
b) does not span V
