LA Lecture 1 Flashcards
What is a typical linear combination of vectors v and w?
cv + dw
What do the combinations c(1,1) + d(2,3) fill?
the whole xy plane
How many solution does these equations have
c + 3d = 0
c + 3d = 0
c +4d = 0
?
No solution, as it’s not on the plane.
What are the components of a column vector v? v= (v1, v2)
V1 = first component, V2 = second component.
How is it called the sum of cv and dw?
a linear combination cv+dw
The vector (2,3) = (2,0)+(0,3) goes across x=___ and up to y=___ in the xy plane.
x=2, y=3
examples of scalar multiplication, vector addition and linear combination
scalar multiplication: cv, dw
vector addition: v+w
linear combination: cv+dw
The combination c(1,1,1) and d(2,3,4) fill _____ in a xyz space.
a plane
0vector + 0vector is called _______
zero vector
0vector + any vector is called _______
any vector in the direction of any vector
v(1,1,1) and w(2,3,5) have _____ dimensions.
3
Unit vector has length of _______
1
What is the length of a vector v?
The square root of v.v.
what is a unit vector
A vector with length one.
What are the standard unit vectors along the x and y axes?
i and j
What is the dot product when vectors are perpendicular?
v.w = 0.
What does a zero dot product indicate in economics?
Balance.
When is the dot product of two vectors zero?
The dot product v.w=0 when the angle between the vectors is 90 degree (right angle)
Why is the zero vector perpendicular to every vector?
The zero vector v=0 is perpendicular to every vector w because 0⋅w=0 is always true.
What is the formula for the cosine of the angle between two nonzero vectors?
cosθ = v.w / ∣∣v∣∣ ∣∣w∣∣
What does the Schwarz Inequality state?
The Schwarz Inequality states:
∣v⋅w∣ ≤ ∣∣v∣∣ ∣∣w∣∣
What does the Triangle Inequality state?
The Triangle Inequality states:
∣∣v+w∣∣ ≤ ∣∣v∣∣ + ∣∣w∣∣
Why is ∣cosθ∣ ≤ 1 for all angles?
For all vector v and w,
∣cosθ∣ ≤ 1, which ensures that: ∣v⋅w∣ ≤ ∣∣v∣∣ ∣∣w∣∣
What does cosθ = 0 imply about the angle θ?
If cosθ = 0, the angle θ between the vectors is 90 degree.
equation of unit vector
u = v / ||v||
what is unit vector U that is in perpendicular to u(1/10, 3/10)
(-3/10, 1/10) or (3/10 or -1/10)
[ swap and negate one ]
(u-v).(u-v)
(u.u)-2(u.v)+(v.v)
The vectors perpendicular to both (1,1,1) and (1,2,3) lie on a _______.
line
All vectors perpendicular to V=(1,1,1) lie on a _____ in 3 dimensions.
plane
If u = (1,1,1) is perpendicular to v and w, then v is parallel to w.
(true/false)
false
If u is perpendicular to v and w, then u is perpendicular to v+2w.
(true/false)
true
If u and v are perpendicular unit vectors, then
||u-v|| = root 2.
true
The solution to 𝐴𝑥=𝑏 can be written as _________.
𝑥=𝐴−1𝑏.
In column picture, when 𝑏 = 0,a combination 𝐴𝑥 of the columns is zero:
one possibility is __________.
𝑥 = (0,….,0)
In row picture, when 𝑏 = 0,all the planes (𝑟𝑜𝑤 𝑖).𝑥 = 0 go through _______.
the center point 𝑥 = (0,….,0)
