Linear Transformation

Question 1

Multiply matrices

Answer 1

With the matrix you are applying the multiplication to, you multiply the first row by the items in the matrix which is being applied. For example, if we have a matrix of:
X
Y
Z
And a 3 x 3 matrix of:
a11 a12 a13
a14 a15 a16
a17 a18 a19
We would apply X to a11, a14 and a17, Y to a12, a15 and a18 and Z to a13, a16, a19.

Question 2

N x N matrix

Answer 2

i = rows, j = columns
So essentially, multiply the kth entry in the ‘i’th row by the k entry in the ‘j’th column, and add all of the multiplications together.

Question 3

Linear Transformation Requirements

Answer 3

1) For any vectors in vector space U, if we add the two vectors and apply the mapping to the result, it will come out as the same vector as mapping to one vector and adding that to he mapping of the other vector. In other words: T(p + q) = T(p) + T(q), with p and q both being vectors in U.
2) If we take any constant value, and take any vector in U, then the result of applying the transformation to constant x vector is the same as multiplying the constant by the transformation of p. In other words: T(ap) = aT(p), where a is a constant.

Question 4

Graphical Effects

Answer 4

Translation -> changing the position
Angular Reflection -> angle of reflection, used in lighting and angle trajectory etc.
Scaling -> the effect of making something smaller of larger