Mathematics fundamentals
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A matrix is an arrangement of numbers organized into rows and columns. It has a specific number of rows and columns called its dimensions. Each individual number in the matrix is called an element, which has a specific location defined by its row and column. A square matrix has the same number of rows and columns. Matrices are useful in computer graphics for transformations and are typically denoted with a capital letter indicating their dimensions. To be equal, two matrices must have the same dimensions and corresponding elements must be identical.
![• In other words, say that An x m = [ai j] and that
Bp x q = [bi j]
• Then A = B if and only if n=p, m=q, and ai j =bi j
for all I and j in range](https://image.slidesharecdn.com/mathematicsfundamentals-140917132806-phpapp01/85/Mathematics-fundamentals-13-320.jpg)
Mathematics fundamentals
- 2. Matrix
- 3. Matrix • A matrix is a collection of numbers arranged into fixed number of rows and columns • C R
- 4. • Each number that makes up a matrix is called an element of the matrix. • The element in a matrix have specific location
- 5. • The numbers of rows and columns of a matrix are called its dimensions • Here it is 4x3
- 6. • 2x3
- 7. Squire matrix • A matrix with the number of rows==columns • In CG squire matrices are used for transformation
- 8. Row matrix • column Matrix
- 9. Notation • Usually a capital letter in bold face like • A or M • Sometimes as a reminder the dimensions are written to the right of • the letter as in B3x3
- 11. • If 2 matrices contain the same No. as elements; are the 2 matrices equal to each other???
- 12. • If 2 matrices contain the same No. as elements; are the 2 matrices equal to each other??? • No, to be equal, must have the same dimensions and must have the same values in the same positions.
- 13. • In other words, say that An x m = [ai j] and that Bp x q = [bi j] • Then A = B if and only if n=p, m=q, and ai j =bi j for all I and j in range
- 14. Zero matrix • Which has all its elements zero •
- 15. Adding • The sum A+B of two m-by-n matrices A and B is calculated entry wise: • (A + B)i,j = Ai,j + Bi,j, where 1 ≤ i≤ m and 1 ≤ j ≤ n.
- 16. • Do you think that • (A + B) +C = A + (B + C)
- 17. • Do you think that • (A + B) +C = A + (B + C) • Yes
- 18. Scalar multiplication • The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:(cA)i,j = c · Ai,j.
- 19. Transpose • The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:(AT)i,j = Aj,i.
- 20. • Familiar properties of numbers extend to these operations of matrices • for example, addition is commutative, i.e., the matrix sum does not depend on the order of the summands: A + B = B + A. • The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and • (A + B)T = AT + BT. • Finally, (AT)T = A.
- 21. • The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.
- 22. • It is called identity matrix because multiplication with it leaves a matrix unchanged: MIn = ImM = M for any m-by-n matrix M.
- 23. Vector • Row matrix • column Matrix
- 24. Vector • Magnitude and the direction of two connecting points in coordinate system is V • If P1 = (x1, y1,z1) is Is the starting point and P2=(x2,y2,z2) is the ending point, then the vector V = (x2-x1, y2-y1, z2-z2)
- 25. Projection in 2D • Projection of v onto the x-axis
- 26. Projection in 3D • Projection of v onto the xz plan
- 27. • The magnitude (length) of a vector : Derived from the Pythagorean theorem – The direction of the vector: α is angular displacment from the x-axis α
- 28. 3D • The magnitude is simple extension of 2D • Direction: • Needs 2 angles to fully describe directions • Latitude/longitude is a real word example
- 29. • α, β, y are the positive angles that the vector makes with each of the positive cordinate axes x,y and z respectivly
- 30. Normalizing • Shrinking or stretching it so its magnitude is 1 – Creating unit vector – Does not change the direction • Normalize by dividing on its magnitude:
- 31. • It doesn’t come out to exactly 1, this is bcz of the error using only 2 decimal places