3- Vectors and Linear Combinations

Question 1

What is a vector?

• What is R^n? How is it usually written?
• What is R^2? And so…?
• What is R^3? And so…?

Answer 1

• it is an ordered pair of n values.
• R^n is the set of all the n-vectors over R, usually written as a nx1 matrices.
• R^2 is the set of all vectors with 2 entries, it represents all points in the plane.
• R^3 is the set of all vectors with 3 entries, I represent all points in the 3d space.

Question 2

• Vector addition;
• scalar multiplication.

Answer 2

• addition of the corresponding entries of xvector and yvector.
• multiplication of each entry in vector v by the scalar alfa.

Question 3

Linear combination of vector v and w:
- what does it represent?
- notation

Answer 3

• the vector obtained by adding two or more vectors which are multiplied by scalar values (that can also be 0)
  -av(vector) + bw(vector)

Question 4

What is a span of vector v and vector w?
And so…

What happen if we are in R^2 and we use 2 linearly independent vectors?

Answer 4

A span is a set of all the linear combinations between vectors v and w.
All the possible vectors that you can reach using vector addition and scalar multiplication.

We can access any vector in the 2d space.

Question 5

Linearly dependent vectors:
- when?
- what does it mean? What do we have?

Answer 5

• when one vector can be expressed as a linear combination of the others: u= av. * bw.
• it means that we have vectors that do not add dimensions to the span and so they can be removed without losing information, we have redundant vectors.

Question 6

Linearly independent vectors:
- when?
- what does it mean?

Answer 6

• a vector can not be expressed as the linear combination of the others.
• each vector adds another dimension to the span (new informations).

Question 7

What is a basis of the vector space?

Answer 7

Set of linearly independent vectors that span the full space.

Question 8

Dot product on vectors:
- also called
- what is it?
- notation
-example

Answer 8

• scalar product
• the mathematical operation that takes 2 vectors and returns a scalar.
• v. * w. = v1w1 + … + vnwn
• (1,2) * (3,4) = 13 + 24 = 3+8= 11

Question 9

L1 and L2 Norms:
- how are also called L1 and L2?
- What is L1 and its notation.
- What is L2 and its notation.

Answer 9

• L1 is the Manhattan distance and L2 is the Euclidean distance.
• L1 is the sum of the absolute values of the vector’s components.
  ||v.||1 = |v1| + … + |vn|
• L2 is the square root of the sum of the squares of the vector’s components.
  ||v.||2 = sqrt( v1^2 + … + vn^2)

Question 10

Line segments between 2 vectors:
- what is it? notation?

Answer 10

• it is the set of points between the vectors:
  { alfav. + betaw. :
  alfa, beta belong to R;
  between 0 and 1;
  and alfa+beta = 1 }