Lecture1

stuff

numbers will also be called scalars
vectors are denoted with lower case
Zero vecotr vector of multiple 0
ones vector vector of multiple 1
e_i vector 0 with a singular 1 somewhere? vector with vectors within is called a list of vectors # could also be a matrix

vectors of the same length can be added to each other making a new vector with added together values.

import numpy as np

vector1 = np.array([1, 2, 3])
vector2 = np.array([4, 5, 6])

result = vector1 + vector2

print(result)

# Output:
# [5 7 9]

vectors of the same length can be substracted to each other making a new vector with substracted together values.

import numpy as np

vector1 = np.array([1, 2, 3])
vector2 = np.array([4, 5, 6])

result = vector1 - vector2

print(result)

# Output:
# [-3 -3 -3]

you can multiple vector with a scalar.

import numpy as np

vector = np.array([1, 2, 3])
scalar = 2

result = vector * scalar

print(result)

# Output:
# [2 4 6]

Here is an example of a linear combination of numpy vectors in Python:

import numpy as np

vector1 = np.array([1, 2, 3])
vector2 = np.array([4, 5, 6])

coeff1 = 2
coeff2 = 3

result = coeff1 * vector1 + coeff2 * vector2

print(result)

# Output:
# [10 21 36]

In this example, two vectors vector1 and vector2 are combined linearly using two scalar coefficients coeff1 and coeff2. The result is a new vector result with the same shape as the input vectors, equal to the sum of the scaled input vectors. The linear combination can be written mathematically as result = coeff1 * vector1 + coeff2 * vector2.

we need product thingy written down

we can try to make sense of this in exercises

same with "differencing" and "sum oand average"

superposition property else known as linear functions

means that you can move fx addition outside of the function and still get the same

a linear function to which you add a constant is called affine. affine functions do not cross 0x and 0y but linear does affine functions are linear functions offset