Quiz 2 Flashcards
What are the conditions that need to be fulfilled for something to be considered a vector space?
Define a set of objects V = {u, v, w, …}. It is a vector space if
Addition
(A1) Closure under addition: If u and v belong to V then so too does u ⊕ v.
(A2) Commutative law for addition: If u and v belong to V, then u ⊕ v = v ⊕ u.
(A3) Associative law for addition: If u, v, and w belong to V, then u ⊕ (v ⊕ w) = (u ⊕ v) ⊕ w.
(A4) There exists a zero vector in V denoted by 0 such that for every vector u in V, u ⊕ 0 = u.
(A5) For every vector u in V there exists a vector –u, called the additive inverse of u, such that u ⊕ –u = 0.
Multiplication
(S1) Closure under scalar multiplication: If u belongs to V, then so too does α ⊙ u for any scalar α.
(S2) For any two scalars α and β and any vector u in V, α ⊙ (β ⊙ u) = (αβ) ⊙ u.
(S3) For any vector u in V, 1 ⊙ u = u.
(S4) For any two scalars α and β and any vector u in V, (α + β) ⊙ u = (α ⊙ u) ⊕ (β ⊙ u).
(S5) For any scalar α and any two vectors u and v in V, α ⊙ (u ⊕ v) = (α ⊙ u) ⊕ (α ⊙ v).
What do you need to do when working with vectors?
Surround + and . with a circle to make ⊕ and ⊙ for addition and multiplication. It signifies that the addition and multiplication may be non-normal
What is a tuple and an n-tuple?
A tuple is a 1 x n row matrix which denotes a point in a R^n space. n is the dimension of the space.
What chapter did you not make notes on?
2.1 in Bronson and thus I should look at to ensure I am confident
What is the magnitude of a tuple?
Define a tuple n = [a_1 a_2 … a_n]
The magnitude is
m = sqrt((a_1)^2 + (a_2)^2 + (a_n)^2))
When is a n-tuple normalized?
When its magnitude is equal to 1
What is ∈ and | in set notation?
∈ means belongs to and | means such that
What is a subspace and how can you use it to cut down the number of steps in proving a set of objects is a vector space?
A subspace is a vector space which is completely contained by another vector space. An example of this might be to define a set
S = [{x y x} ∈ R^3 | y = 0]
This obviously is a subset of the vector space R^3 which is already proved to be a vector space.
Due to this knowledge, we now only have to prove 3 properties
Closure under addition: Define u, v ∈ S, then u + v ∈ S
Closure under multiplication: Define u ∈ S and also the variable a which is some arbitrary scalar. If these are both true then a * u ∈ S
If you can prove both of these to be true then S is a vector space
That the zero vector for the larger set is also contained within the subset (So for the example provided [0 0 0] is the zero vector for R^3, and so for S to be a subset [0 0 0] must also be contained in S)
What does it mean for u to be a linear combination of vectors v w and x?
It means that for some scalars a, b, and c
u = av + bw + cx
What is the span of a set of vectors?
It is the set of all linear combinations of those vectors
When is a set of vectors linearly independent?
When the only solution to c1v1 + c2v2 + … + cnvn = 0 is
c1 = c2 = … = cn = 0
