Change of basis from RGB to different set of colors Color images are defined by red, green, blue (RGB) values at each pixel.
What if instead I wanted to define pixel colors by orange, black, and yellow? Is there a way to change the basis of colors from RBG to OKY (orange, black, yellow) in order to define my pixels in terms of these new colors?
Orange, Black, and Yellow technically "Span" the same "colorspace" as Red, blue, and green, so I should be able to recreate colored images with these 3 colors right? How could I calculate the conversions from RGB to OKY.
I have a image that is predominately composed of black, orange, and yellow colors. Therefore black, orange, and yellow is a more natural "color basis" for this picture than Red Blue and Green. 
So how would I figure out the conversions?
 A: I think this might qualify as an "X/Y" question scenario.
The question has two upvotes, so I will answer — however I feel I need to lay some foundations of color theory first as the question as asked has only an "imaginary" answer — the question itself indicates a misunderstanding of color perception. 
Also I am not certain this stack-subject is the best for this question, however I do hope my answer makes it a useful combination.
BASIC COLOR THEORY
Human color vision uses both tristimulus and opponent models of color. All color theory is essentially concerned with how those psychophysical aspects of vision are stimulated.
Additional useful Google terms here are CIEXYZ 1931, CIELAB, CMYK, and RGB.
TRISTIMULUS COLORS
The human eye has 4 basic types of photo receptors. 


*

*"Rods" which are sensitive to dark conditions (Scotopic), and are not substantially involved in color vision. 

*L, M, S "Cones" which are sensitive to Red, Green, and Blue
wavelengths of light (respectively) in normal day lighting conditions
(Photopic). 


Cones sense the range of visible light with sensitivity varying based on wavelength. L is sensitive to the longest visible wavelengths, which we identify as red, M is sensitive to a middle wavelength identified as green, and S to the shortest visible wavelengths, identified as blue or violet.
All color perception begins by stimulating these three cone types in some ratio to achieve a particular color perception. This leads to a general understanding that Red, Green, and Blue make up the "primary" colors of light. (however the reality is more complicated, "primary colors" are a convenient fiction).
OPPONENT COLORS
Vision processing starts in the eye, after sensing the RGB tristimulus values, the next stage involves the ganglion cells which matrix the RGB into three opponent channels of dark-to-light, red/green, and blue/yellow. This encoding in the eye reduces the total data sent over the optic nerve to the brain for further processing.
Red/Green are opponent colors — you can't see a "redish green" color. When the L and M cones are stimulated about equally either by spectral yellow, or a mixture of red and green light, then you will perceive yellow, not a greenish red.
Blue/Yellow are opponent colors — You can't see a blueish yellow. When mixed, B/Y makes grey or white because all three cones are stimulated in that mix.
Thus the opponent colors create the "four unique colors" of red green yellow blue.
COLOR MODELS AND SPACES
RGB is an additive color model. Adding various amounts of red, green, and/or blue light will create many other colors. 
CMYK is a subtractive color model. The inks of your printer absorb (subtract) light, and reflect specific wavelengths to create color sensations. 
A colorspace is a definition of a color model that defines the "primaries" white point, gamma, etc.
ALL IN YOUR HEAD
Color is all in your mind. In nature, there are simply various wavelengths of light. It is your eye/brain that processes those various wavelengths into the perception of color.
Because your eye is sensitive to three basic colors (tristimulus) you can see non-spectral colors, that is colors that do not exist as a single wavelength. 
Purple for instance is non-spectral — purple does not exist in reality, it exists only because your red and blue cones are being stimulated but not the green cone.

Your computer monitor is additive, and uses the color "primaries" of red, green, blue. Each are chosen to "mostly stimulate" each of the L, M, and S cones as independently as possible. So the RGB spectral values in the monitor are not at the peaks of each of the cone's sensitivities, but offset to affect one cone and not the others to the degree possible).
Imaginary Colors
In the intro, I indicate the original question only had "imaginary" answers — I was alluding to some colorspaces such as XYZ which use "imaginary" primaries. These are primaries that are in the color space for mathematical convenience, but that otherwise do not exist in reality.
ANSWERING YOUR QUESTIONS

a way to change the basis of colors from RBG to OKY

By "basis" I think you mean "color model". Orange, black, and yellow is not a useful color model for an emissive computer monitor.
Orange, Black, and Yellow can certainly be used for printed media, and is (as well as other highly saturated colors, as Pantone spot colors for instance). This is done by assigning various spot colors (eg Pantone) to the various graphical elements. 
Using spot colors is often done for colors that don't print well with the simple CMYK process. You might use CMYK plus various spot colors, to get a more vibrant design for example. or a set of spot colors by themselves. You can set these in Photoshop or Illustrator.
And of course there are print processes that use more than just CMYK, such as Hexachrome. 

in order to define my pixels in terms of these new colors?

Your computer monitor is locked into red green and blue as a matter of the hardware. You can't change that — you'd have to design and engineer a whole new monitor and there is no point in doing so for your example.
For that matter, your CMYK printer is "locked in" too — though I suppose you could create a custom ink cartridge using O,Y,K inks and then create a custom profile based on printing with that cartridge (this requires a spectrophotometer and profiling software).

Orange, Black, and Yellow technically "Span" the same "colorspace" as Red, blue, and green,

No they do not, and it's not even close. 

so I should be able to recreate colored images with these 3 colors right? 

No, absolutely NOT. Your images will be limited to tones and shades of the colors between orange and yellow, which is a much smaller gamut that the gamut of CMYK.

How could I calculate the conversions from RGB to OKY.

Such conversions could be done with a matrix, or a LUT (look up table).

I have a image that is predominately composed of black, orange, and
  yellow colors. Therefore black, orange, and yellow is a more natural
  "color basis" for this picture than Red Blue and Green.

No. If you are talking about your monitor, what's "natural" is a set of tristimulus primary colors that stimulate each of the cones to create the color sensations.
For printing with inks, then yes, using certain spot colors may yield a better result than plain CMYK — but probably not for your specific use case. CMYK already has yellow and black. Instead of orange it has magenta which makes orange when mixed with yellow.
Spot colors are usually used for colors that CMYK has a hard time creating, like saturated red.

So how would I figure out the conversions?

I would suggest looking at the following:
Bruce Lindbloom has a lot of math online for various conversions.
Charles Poynton's Color FAQ is also useful.
And the Handprint Color Theory pages are excellent.
And read up on CIEXYZ and CIELAB for more on these "standard observer" spaces.
A: RGB can be easily converted to XYZ values using the the matrix of the known constants. This isn't the case with OKY. 
So if both the consumer and supplier have the matrix which converts OKY to XYZ  then this system is fine to use. Otherwise an ambiguity might be generated leading to generation of wrong colors. More precisely observer metamerism(assuming same light souce/illuminant or other conditions)The user might want a particular color while the producer may produce something else. XYZ values are tristimulus values defined by the CIE 1931 system. Thus conclusion is that if you have the conversion matrix from $OKY$ to $XYZ$ and the same one is used by your counterpart. Thus you can use the path $OKY->XYZ->RGB$ http://www.brucelindbloom.com/index.html?Eqn_RGB_XYZ_Matrix.html see this link.
