SciTech-Math-AdvancedAlgebra-Linear Spaces(Vector Spaces) and Subspace: The Column Space of a Matrix

Resources:

• AMS: Open Math Notes
  
  a repository of freely downloadable mathematical works hosted by the American Mathematical Society as a service to researchers, faculty and students.
  
  https://www.ams.org/open-math-notes/files/course-material/OMN-201908-110802-1-Course_notes-v1.pdf

• Vector Spaces and Subspaces:
  
  https://math.mit.edu/~gs/dela/dela_5-1.pdf
  
  https://web.mit.edu/18.06/www/: 18.06 Linear Algebra@MIT
  
  https://math.mit.edu/~gs/: Gilbert Strang

• Linear Algebra and Vector Analysis:
  
  https://people.math.harvard.edu/; Math 22b Spring 2019: https://people.math.harvard.edu/~knill/teaching/math22b2019/
  
  Math 22b Spring 2019, 22b Linear Algebra and Vector Analysis
  
  

• Vector spaces, operators and matrices
  
  https://web.stanford.edu/group/dabmgroup/cgi-bin/dabm/wp-content/uploads/2021/12/Lecture_14.pdf

• CMU: Linear Spaces
  
  https://www.math.cmu.edu/~wn0g/noll/2ch1a.pdf

• Department of Mathematics at Columbia University
  
  https://www.math.columbia.edu/department/pinkham/HCP_LinearAlgebra.pdf

N-element ordered Number Sequence

Sets: K^n, Operations:{ScalarMultiplication, Addition, Other8rules}

Vector: N-Dimensional vector

Relationship between Vectors:

• Linear Combinations
• Linear Representation
• Linear
  
  Linear Spaces(Vector Spaces) are Sets
  
  Linear Combinations

import cv2 as cv

img_garden = cv.imread("/Users/abaelhe/Desktop/TheGarden.png")
img_emily = cv.imread("/Users/abaelhe/Desktop/EmilyDickinson.png")
math_merged = (
  ratio*img_garden + (1.0-ratio)* img_emily
).astype(np.uint8)

img_garden_gray = cv.cvtColor(img_garden, cv.COLOR_RGB2GRAY)
img_emily_gray = cv.cvtColor(img_emily, cv.COLOR_RGB2GRAY)
math_merged_gray = (
  ratio*img_garden_gray + (1.0-ratio)* img_emily_gray
).astype(np.uint8)

def merge_flatten_img(
  background, front,
  bg_ratio=0.4, filter_ratio=0.3):
    front_mean = front.mean()
    front_min = int(front_mean*(1-filter_ratio))
    front_max = int(front_mean*(1+filter_ratio))
    ret = []
    for (bg, ft) in zip(background, front):
        if (front_min <= ft <= front_max):
            ret.append( int(bg_ratio*bg + (1-bg_ratio)*ft) )
        else:
            ret.append( ft )
    return np.array(ret, dtype=np.uint8)

math_filter_gray = merge_flatten_img(
  img_garden_gray.flatten(), img_emily_gray.flatten()
).reshape(img_emily.shape[:-1])

im_r = merge_flatten_img(
  img_garden[:,:,0].flatten(), img_emily[:,:,0].flatten()
)
im_g = merge_flatten_img(
  img_garden[:,:,1].flatten(), img_emily[:,:,1].flatten()
)
im_b = merge_flatten_img(
  img_garden[:,:,2].flatten(), img_emily[:,:,2].flatten()
)
im_zip = tuple(zip(im_r.tolist(), im_g.tolist(), im_b.tolist()))
math_filter_color = np.array(
  im_zip, dtype=np.uint8
).flatten().reshape(img_emily.shape)

imgshow(math_merged, save_to='/Users/abaelhe/Desktop/math_merged.png')
imgshow(math_merged_gray, save_to='/Users/abaelhe/Desktop/math_merged_gray.png')
imgshow(math_filter_gray, save_to='/Users/abaelhe/Desktop/filter_gray.png')
imgshow(math_filter_color, save_to='/Users/abaelhe/Desktop/filter_color.png')

#TheGarden:	#EmilyDickinson:
#math_merged:	#filter_color:
#math_merged_gray:	#filter_gray: