This document discusses how linear combinations can be used to represent colors in additive and subtractive color models. In additive color models like RGB used for screens, colors are represented as vectors where each primary color (red, green, blue) has an intensity value. Any color can be created as a linear combination of these primary vectors by assigning intensity values and adding the result. Similarly in subtractive color models like CMYK for printing, primary colors (cyan, magenta, yellow, black) are represented as vectors that can be linearly combined by assigning concentration values to each and adding the results to produce any color. Examples are given of how specific linear combinations of RGB and CMYK primary vectors can produce

1. Linear Combination In Color Model
Prepared by:
Md. Washim (2015-3-60-032)
Sharmin Sultana Koly (2015-3-65-010)
Sheikh Aminul Islam (2015-3-65-013)

2. Introduction
The project was given as the review of the
course “Linear Algebra and Complex Variable” to
show the output of the course learning and
utilizing. Our project is about to show the use of
Linear combination to solve problems.

3. Linear Combination
In mathematics, a linear combination is an
expression constructed from a set of terms by
multiplying each term by a constant and adding
the results (e.g. a linear combination of u and v
would be any expression of the form k1u + k2v,
where u and v are constants).

4. Color Model
A color model is a system for creating a full
range of colors from a small set of primary
colors. There are two types of color models:
1. Additive color models use light to display
color. Example: RGB colors used in software UI
designing.
2. Subtractive color models uses printing inks.
Example: CMYK colors used in modern printers.

5. RGB (Used in Electronic Media)

6. CMYK (Used in Print Media)

7. RGB (Additive color model)
In computer, RGB color is usually stored as a triplet of
integer numbers in the range 0 to 255.
(255,0,0) is Red
(0, 255,0) is Green
(0,0, 255) is Blue
We can view those colors as the vectors in 3D space
having those components.
Any other color is obtained as a linear combination of
these basic colors.

8. Linear Combination in RGB
Let’s take these three colors as three vectors,
r for Red
g for Green
b for Blue
Then the Linear Combination w is,
w = k1r + k2g + k3b
Where k1, k2, k3 are three scalars.

9. Now, If we take components as, r = (255,0,0), g =
(0,255,0), b = (0,0,255)
For k1 = 0.6, k2 = 0 and k3 = 0.8 the Linear
combination is,
w = 0.6 (255, 0, 0) + 0 (0, 255, 0) + 0.8 (0, 0, 255)
= (153, 0, 204)
This linear combination of 3 colors form Purple color.

11. Linear Combination in CMYK
Let’s take these four colors as four vectors:
c for Cyan
m for Magenta
y for Yellow
k for key (Black)
Then the Linear Combination w is,
w = n1c + n2m + n3y + n4k
Where n1, n2, n3 and n4 are three scalars.

12. Now, If we take components as, c = (100, 0, 0, 0), m =
(0, 100, 0, 0), y = (0, 0, 100, 0), k = (0, 0, 0, 100)
For n1 = 0.5, n2 = 0.3, n3 = 0 and n4 = 0.05 the Linear
combination is,
w = 0.5 (100, 0, 0, 0) + 0.3 (0, 100, 0, 0) + 0 (0, 0, 100,
0) + 0.05 (0, 0, 0, 100)
= (50, 30, 0, 5)
This linear combination of 4 colors form Sky Blue
color.

14. Conclusion
From this course “Linear Algebra and Complex
Variables” we have learnt a lot of things which are used
in our real life in various ways. Doing this project, we
could feel the reality of mathematics and hopefully we
will use the other methods learnt from this course and
try to solve harder problem in our real life.