Hi,
So far, we have covered little bits about vectors, matrices, etc. Now, let's discuss about functions. These are the ideas that later lead us to transformations. So, let's begin!
int sample_function(int num)
{
return num * num;
}
This takes a number, and gives the square of it. Mathematically, this can be written as:$$f(x) = x ^ 2$$
Now, this is a perfectly valid vector valued function. These functions generally do a mapping from$$R^m -> R^n$$
In the above example, it is a mapping from R3 to R2. These functions are called as Transformations. It is just a different name given to the same concept we had for scalar valued functions. The general notation used for transformation is T.
If a transformation obeys the above two points, it is a linear transformation. A linear transformation generally would involve linear combinations of components.
Now, let us take an example. Let $$T(x_1, x_2) = (x_1 + x_2, 3x_1)$$
Now, let $$\vec a = (a_1, a_2) and \vec b = (b_1, b_2)$$ $$T(\vec a) + T(\vec b) = (a_1 + a_2, 3a_1) + (b_1 + b_2, 3b_1) = (a_1 + a_2 + b_1 + b_2, 3a_1 + 3b_1)$$
Now,$$T(\vec a + \vec b) = (a_1 + b_1 + a_2 + b_2, 3(a_1 + b_1)$$
Clearly, both these are same. Similarly, we can prove that $$T(c\vec a) = cT(\vec a)$$
Thus, the above transformation is a linear transformation. Now, it can be shown that $$T(x_1, x_2) = (x_1^2, 0)$$ is a non-linear transformation. So, generally, transformations would be linear if they have linear combinations of components.
So far, we have covered little bits about vectors, matrices, etc. Now, let's discuss about functions. These are the ideas that later lead us to transformations. So, let's begin!
What are functions?
In simple terms, Functions are basically a mapping from one set to another. If we take examples like sin, cos, log, etc, these are basically functions that take an input and give an output. So, it takes a number and gives a number. And if you're familiar with computer programming, here is what a C function looks like:int sample_function(int num)
{
return num * num;
}
This takes a number, and gives the square of it. Mathematically, this can be written as:$$f(x) = x ^ 2$$
Scalar valued and Vector valued functions:
Functions like sin, cos and the above example are all called as scalar valued functions. What this means is, these functions return a value in R. So, the output is generally a real number. But, functions are not restricted to just this. Functions can be what we call as vector valued functions. These are functions that return tuples of Rn. For example: $$f(x_1, x_2, x_3) = (x_1 + x_2, x_1 + x_3)$$Now, this is a perfectly valid vector valued function. These functions generally do a mapping from$$R^m -> R^n$$
In the above example, it is a mapping from R3 to R2. These functions are called as Transformations. It is just a different name given to the same concept we had for scalar valued functions. The general notation used for transformation is T.
Linear Transformations
A linear transformation is a function(or transformation) iff $$T(\vec a + \vec b) = T(\vec a) + T(\vec b)$$ $$T(c\vec a) = cT(\vec a)$$If a transformation obeys the above two points, it is a linear transformation. A linear transformation generally would involve linear combinations of components.
Now, let us take an example. Let $$T(x_1, x_2) = (x_1 + x_2, 3x_1)$$
Now, let $$\vec a = (a_1, a_2) and \vec b = (b_1, b_2)$$ $$T(\vec a) + T(\vec b) = (a_1 + a_2, 3a_1) + (b_1 + b_2, 3b_1) = (a_1 + a_2 + b_1 + b_2, 3a_1 + 3b_1)$$
Now,$$T(\vec a + \vec b) = (a_1 + b_1 + a_2 + b_2, 3(a_1 + b_1)$$
Clearly, both these are same. Similarly, we can prove that $$T(c\vec a) = cT(\vec a)$$
Thus, the above transformation is a linear transformation. Now, it can be shown that $$T(x_1, x_2) = (x_1^2, 0)$$ is a non-linear transformation. So, generally, transformations would be linear if they have linear combinations of components.
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