0 - Special Topic: Gram-Schmidt Process

Question 2

What does normalizing a vector do, and how do we calculate it?

Answer 2

Normalizing a vector shrinks or scales it to its unit vector form. This is done by normalizing or dividing the vector’s components by its magnitude. The magnitude is found by taking the square root of the squares of the components.

ex) u₁ = v₁ / ||v₁||

= v₁ / sqrt( a² + b² )

…where a and b are vector v’s components

Question 3

Define the “dot” product”.

Answer 3

Dot Product = “Scalar” Product

= “Sum of Products”

= component-wise multiplication, then add them all up

Question 4

How do you calculate a…

…proj_b(a) = projection onto b of a?

…proj_a(b) = projection onto a of b?

Answer 4

proj_b(a) = projection onto b of a​

= some scaled value of the vector b

= ( scalar ) x vector b

= (dot product of a • b) x vector b

= (a • b) x b

…and the reverse for proj_a(b)

Question 5

What is the Gram-Schmidt Process in math terms?

Answer 5

The Gram-Schmidt Process is a method of turning a given set of vectors into a set of orthonormal vectors that spans the original entire set.

You can think of it as a 3 part process:

1. Pick any arbitrary vector to be our first vector and then normalize that vector by its magnitude. This will give us our first unit vector, u₁
   u₁ = v₁ / ||v₁||
2. Then take our next vector and subtract out its projection onto the first vector of this second vector. Then normalize this vector by its magnitude.
   u₂ = ( v₂ - proj_u1(v₂) ) / ||v₂||
3. Take our next vector and subtract out its projections onto the first and second vectors. Then normalize this by its magnitude.
   u₃ = ( v₃ - proj_u1(v₃) - proj_u2(v₃) ) / ||v₃||
4. Repeat step 3 for the rest of the vectors.
   u₄ = ( v₄ - proj_u1(v₄) - proj_u2(v₄) - proj_u3(v₄) - … ) / ||v_{<span>4</span>}||

Question 6

What is the Gram-Schmidt Process ​in words?

Answer 6

The Gram-Schmidt Process is a method of turning a given set of vectors into a set of orthonormal vectors that spans the original entire set.

You can think of it as a 3 part process:

1. Pick any arbitrary vector to be our first vector and then normalize that vector by its magnitude.
2. Then take our next vector and subtract out its projection onto the first vector of this second vector. Then normalize this vector by its magnitude.
3. Take our next vector and subtract out its projections onto the first and second vectors. Then normalize this by its magnitude.
4. Repeat step 3 for the rest of the vectors.

Question 7

When you have two vectors a, b with their tails glued at the origin and pointing in different directions, how do you calculate the vertical component of vector a?

Answer 7

= vertical component of a

= entire vector a - the horizontal component of a

= a - proj_b(a)

Question 8

What does it mean to be orthogonal?

What does it mean to be orthonormal?

Answer 8

Orthogonal = vectors are 90 degrees from each other = perpendicular

Orthonormal = vectors are orthogonal AND normalized

(ie - they are the unit vectors making up the fundamental directions)