# Linear Transformations

When every element undergoes multiplication or increased by a constant it is called as linear transformation. Linear transformation impacts standard deviation, mean, IQR and other crucial numerals in numerous ways.

## Sum of Linear Transformations

Whenever a constant c are summed up to each extremity (member) of a set the mean would be c more than it was prior to the constant was summed. The variance and standard deviation won't be impacted and the IQR won't be impacted. We'll prove this facts now, having μ and σ be the standard deviation & mean, on the other hand, prior to adding c and μt and σt be the standard deviation & mean and after the shift. Lastly, we allow the original set be {a1, a2, . . . , an}, hence the translated set is {a1 + c, a2 + c, . . . , an + c}.

Here we apply the solution of the 1st equation to substitute μt with μ+c in the 2nd equation. As the variance is merely the standard deviation's square, the reality is the standard deviation is not impacted implies that the variance will not be either.

## Linear Transformation Multiplication

A different type of shift is multiplication. If every member of a set by multiplication process by a constant c, and then the mean would be c times its value prior to the constant was multiplied, the standard deviation would be |c| times its value prior to constant undergoes multiplication and IQR would be |c| times its value. By applying the same notation like before, we get:

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