LA Lecture 1 Flashcards

1
Q

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

A

cv + dw

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2
Q

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

A

the whole xy plane

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3
Q

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

A

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

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4
Q

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

A

V1 = first component, V2 = second component.

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5
Q

How is it called the sum of cv and dw?

A

a linear combination cv+dw

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6
Q

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

A

x=2, y=3

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7
Q

examples of scalar multiplication, vector addition and linear combination

A

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

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8
Q

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

A

a plane

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9
Q

0vector + 0vector is called _______

A

zero vector

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10
Q

0vector + any vector is called _______

A

any vector in the direction of any vector

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11
Q

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

A

3

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12
Q

Unit vector has length of _______

A

1

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13
Q

What is the length of a vector v?

A

The square root of v.v.

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14
Q

what is a unit vector

A

A vector with length one.

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15
Q

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

A

i and j

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16
Q

What is the dot product when vectors are perpendicular?

17
Q

What does a zero dot product indicate in economics?

18
Q

When is the dot product of two vectors zero?

A

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

19
Q

Why is the zero vector perpendicular to every vector?

A

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

20
Q

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

A

cosθ = v.w / ∣∣v∣∣ ∣∣w∣∣

21
Q

What does the Schwarz Inequality state?

A

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

22
Q

What does the Triangle Inequality state?

A

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

23
Q

Why is ∣cosθ∣ ≤ 1 for all angles?

A

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

24
Q

What does cosθ = 0 imply about the angle θ?

A

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

25
Q

equation of unit vector

A

u = v / ||v||

26
Q

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

A

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

27
Q

(u-v).(u-v)

A

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

28
Q

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

29
Q

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

30
Q

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

31
Q

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

32
Q

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

33
Q

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

A

𝑥=𝐴−1𝑏.

34
Q

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

A

𝑥 = (0,….,0)

35
Q

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

A

the center point 𝑥 = (0,….,0)