Linear Algebra

Question 1

List the 4 properties of a vector space.

Answer 1

1. There is a “zero vector” in V, appropriately called “0”, so that v + 0 = v for all v∈V.
2. If v is a vector in V, then there is a vector -v∈V so that v + (-v) = 0.
3. Closed under addition: If u and v are vectors in V, then u + v is also in VV.
4. Closed under scalar multiplication: If u is a vector in V, then cu is also in V for any constant c.

Question 2

Suppose that V1 and V2 are both subspaces of V. Which of the following is NOT necessarily a subspace of V?

1. The intersection of V1 and V2.
2. The sum of V1 and V2.
3. The union of V1 and V2.
4. These are all subspaces of V.

Answer 2

The union of V1 and V2 is NOT necessarily a subspace of V.

Proving the sum is a subspace:
- 0 is part of both V1 and V2, so 0 + 0 = 0 ∈ V1 ⊕ V2.
​- Addition: Suppose v1, v2 ∈ V1 ⊕ V2, and write v1 = a1 + b1, v2 = a2 + b2 where a1,a2 ∈ V1, and b1,b2 ∈ V2. Then v1 + v2 = a1 + b1 + a2 + b2 = (a1 + a2) + (b1 + b2). But then a1 + a2 ∈ V1 and b1 + b2 ∈ V2, so (a1 + a2) + (b1 + b2) ∈ V1 ⊕ V2.
- A similar process works for scalar multiplication.

Counter example for union:
If we let V be the set of real ordered pairs, V1 be the set of pairs (x,0), and V2 be the set of pairs (0,x). Then V1 ∪ V2 is missing pairs like (1,1), so it is not a vector space as (1,0) and (0,1) are in the set but their sum is not.

Question 3

If V and W are subspaces of the same vector space, then what can we tell about their dimensions?

Answer 3

dim(V+W) + dim(V∩W) = dim(V) + dim(W)

Question 4

If a, b are real-valued vectors, give the 2 equivalent definitions of the dot product a⋅b?

Answer 4

The result is a single real number.

Definition 1:
a⋅b = ‖a‖⋅‖b‖⋅cosθ
where
‖a‖ - the norm (length) of a.
θ - angle between the two vectors.

Definition 2:
If a = (a_1, a_2, …, a_n) and b = (b_1, b_2, …, b_n), then:
a⋅b = a_1⋅b_1 + a_2⋅b_2 + … + a_n⋅b_n

Question 5

How we define the norm of a vector a = (a1, a2, …, an)?

Answer 5

‖a‖=‖(a1, a2, …, an)‖=sqrt(a1^2, a2^2, …, an^2)

Question 6

What is Cauchy-Schwarz Inequality? Give the

Answer 6

x⋅y ≤ ‖x‖⋅‖y‖

Question 7

What is the identity matrix of nxn?

Answer 7

The matrix I where all elements are zeros except the diagonal, where all elements are ones.

The identity matrix satisfies: AI = IA = A

Question 8

How do we define matrix multiplication?

Answer 8

AB = C

So C(i,j) will be the dot-product of row i in A and column j in B.

Matrix multiplication is NOT commutative, meaning it is generally not true that AB = BA. If it is, we say AB are commute.

Question 9

What is the trace of a matrix (a square one)?

Answer 9

The sum of entries in the main diagonal.

tr(A) = tr(A^T)

Question 10

What are the elementary reduction operations on systems that keep the set of solutions?

Answer 10

1. Swapping between equations
2. Scalar multiplication on both sides of an equation
3. Row combination - an equation is replaced by the sum of of itself and a multiple of another

Question 11

Define a row echelon form

Answer 11

In each row of a system, the first variable with a nonzero coefficient is the row’s leading variable.

A system is in echelon form if each leading variable is to the right of the leading variable in the row above it, except for the leading variable in the first row, and any all-zero rows are at the bottom.

Question 12

Define an homogeneous linear equation

Answer 12

A linear equation is homogeneous if it has a constant of zero, so that it can be written as a1x1+a2x2+…+anxn=0

Question 13

If V is a vector space and S ⊆ V, then what makes S a subspace?

Answer 13

S is a subspace of V if S is closed under linear combinations of pairs of vectors:
For any vectors s1,s2 in S and scalars r1,r2, the vector r1s1+r2s2 is in S

Another way - S is a subspace if and only if it is a span

Question 14

For a non-empty subset S of a vector space, define Span(s)

Answer 14

Span(S) is the set of all linear combinations of vectors from S.

Span(S) = { c1s1 + … + cnsn | c1,…,cn in R and s1,…,sn in S }

Question 15

What is the connection between a spanning set, linearly independent set and a basis?

Answer 15

A spanning set is minimal when it is linearly independent.

A linearly independent set is maximal when it spans the space.

A sequence of vectors that is linearly independent AND spans that space called a basis.

Question 16

Define the row space of a matrix, together with row rank.

Answer 16

The row space of a matrix is the span of the set of its rows. The row rank is the dimension of this space - the number of linearly independent rows.