Let us try to understand a Julia vector [commented code]
julia> using LinearAlgebra
julia> x = [1,2,3] #x is a Julia vector
3-element Vector{Int64}:
1
2
3
julia> size(x)
(3,)
julia> A = rand(3,3)
3×3 Matrix{Float64}:
0.46205 0.907979 0.57372
0.0121377 0.590446 0.376092
0.672652 0.909502 0.304637
julia> A*x #x acts as a column vecotor
3-element Vector{Float64}:
3.999167421183575
2.3213037773354586
3.4055688127482697
julia> dot(A[1,:],x) #First element of A*x is the dot product between first row of A and x
3.999167421183575
julia> x*A #Julia thinks x is a matrix with dimension (3,1)
ERROR: DimensionMismatch: matrix A has dimensions (3,1), matrix B has dimensions (3,3)
Let us try to make out the behaviour of a vector in python/numpy
>>> import numpy as np
>>> x = np.array([1,2,3])
>>> x.shape
(3,)
#x is an orientationless array
>>> A = np.random.random((3,3)) #A is a 3 x 3 matrix
>>> A
array([[0.62256804, 0.48535832, 0.86000692],
[0.24275571, 0.22393731, 0.5531013 ],
[0.54999826, 0.90498169, 0.35720945]])
>>> A@x #Left mulitply x with matrix A
array([4.17330543, 2.34993421, 3.43158999]) #x acts as a column vector
>>> np.dot(A[0,:],x) #First element of A@x is the dot product between first row of A and x
4.173305432498873
>>> x@A
array([2.75807425, 3.64817801, 3.03783785])
>>> np.dot(A[:,0],x)
2.7580742486325107 #First element of x@A is the dot product between first column of A and x
>>>
Julia seems to be very particular about the orientation of julia vector while defining.
But Python treats a vector as both column and row vector.
A row vector in Julia can be made as
x = [1 2 3] #without commas in between elements
1×3 Matrix{Int64}:
1 2 3
A row vector in Python can be made as
x = np.array([[1, 2, 3]])
A column vector in Python can be made as
x = np.array([[1], [2], [3]])
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