LA Lecture 1

Question 1

What is a typical linear combination of vectors v and w?

Answer 1

cv + dw

Question 2

What do the combinations c(1,1) + d(2,3) fill?

Answer 2

the whole xy plane

Question 3

How many solution does these equations have
c + 3d = 0
c + 3d = 0
c +4d = 0
?

Answer 3

No solution, as it’s not on the plane.

Question 4

What are the components of a column vector v? v= (v1, v2)

Answer 4

V1 = first component, V2 = second component.

Question 5

How is it called the sum of cv and dw?

Answer 5

a linear combination cv+dw

Question 6

The vector (2,3) = (2,0)+(0,3) goes across x=___ and up to y=___ in the xy plane.

Answer 6

x=2, y=3

Question 7

examples of scalar multiplication, vector addition and linear combination

Answer 7

scalar multiplication: cv, dw
vector addition: v+w
linear combination: cv+dw

Question 8

The combination c(1,1,1) and d(2,3,4) fill _____ in a xyz space.

Answer 8

a plane

Question 9

0vector + 0vector is called _______

Answer 9

zero vector

Question 10

0vector + any vector is called _______

Answer 10

any vector in the direction of any vector

Question 11

v(1,1,1) and w(2,3,5) have _____ dimensions.

Answer 11

3

Question 12

Unit vector has length of _______

Answer 12

1

Question 13

What is the length of a vector v?

Answer 13

The square root of v.v.

Question 14

what is a unit vector

Answer 14

A vector with length one.

Question 15

What are the standard unit vectors along the x and y axes?

Answer 15

i and j

Question 16

What is the dot product when vectors are perpendicular?

Answer 16

v.w = 0.

Question 17

What does a zero dot product indicate in economics?

Answer 17

Balance.

Question 18

When is the dot product of two vectors zero?

Answer 18

The dot product v.w=0 when the angle between the vectors is 90 degree (right angle)

Question 19

Why is the zero vector perpendicular to every vector?

Answer 19

The zero vector v=0 is perpendicular to every vector w because 0⋅w=0 is always true.

Question 20

What is the formula for the cosine of the angle between two nonzero vectors?

Answer 20

cosθ = v.w / ∣∣v∣∣ ∣∣w∣∣
​

Question 21

What does the Schwarz Inequality state?

Answer 21

The Schwarz Inequality states:
∣v⋅w∣ ≤ ∣∣v∣∣ ∣∣w∣∣

Question 22

What does the Triangle Inequality state?

Answer 22

The Triangle Inequality states:
∣∣v+w∣∣ ≤ ∣∣v∣∣ + ∣∣w∣∣

Question 23

Why is ∣cosθ∣ ≤ 1 for all angles?

Answer 23

For all vector v and w,
∣cosθ∣ ≤ 1, which ensures that: ∣v⋅w∣ ≤ ∣∣v∣∣ ∣∣w∣∣

Question 24

What does cosθ = 0 imply about the angle θ?

Answer 24

If cosθ = 0, the angle θ between the vectors is 90 degree.

Question 25

equation of unit vector

Answer 25

u = v / ||v||

Question 26

what is unit vector U that is in perpendicular to u(1/10, 3/10)

Answer 26

(-3/10, 1/10) or (3/10 or -1/10) [ swap and negate one ]

Question 27

(u-v).(u-v)

Answer 27

(u.u)-2(u.v)+(v.v)

Question 28

The vectors perpendicular to both (1,1,1) and (1,2,3) lie on a _______.

Answer 28

line

Question 29

All vectors perpendicular to V=(1,1,1) lie on a _____ in 3 dimensions.

Answer 29

plane

Question 30

If u = (1,1,1) is perpendicular to v and w, then v is parallel to w. (true/false)

Answer 30

false

Question 31

If u is perpendicular to v and w, then u is perpendicular to v+2w. (true/false)

Answer 31

true

Question 32

If u and v are perpendicular unit vectors, then ||u-v|| = root 2.

Answer 32

true

Question 33

The solution to 𝐴𝑥=𝑏 can be written as _________.

Answer 33

𝑥=𝐴−1𝑏.

Question 34

In column picture, when 𝑏 = 0,a combination 𝐴𝑥 of the columns is zero: one possibility is __________.

Answer 34

𝑥 = (0,….,0)

Question 35

In row picture, when 𝑏 = 0,all the planes (𝑟𝑜𝑤 𝑖).𝑥 = 0 go through _______.

Answer 35

the center point 𝑥 = (0,….,0)