# Linear mapping

### Linear mapping

Let V and W are the vector spaces over field K. A function f: V-> W is said to be the linear map for two vector v,u \ and a scalar c \ K:

- If the transformation is additive in nature:

- If they are multiplicative in nature in terms of a scalar.

### Zero/Identity Transformation

A linear transformation from a vector space into itself is called Linear operator:

**Zero-Transformation:**For a transformation is called zero-transformation if:

**Identity-Transformation:**For a transformation is called identity-transformation if:

### Properties of Linear Transformation

Let T: V \rightarrow W be the linear transformation where u,v \epsilon V. Then, the following properties are true:

- If then,

## Linear Transformation of Matrix

Let T be a mxn matrix, the transformation T: is linear transformation if:

**Zero and Identity Matrix operations**

- A matrix mxn matrix is a
*zero matrix*, corresponds to zero transformation from R^n \rightarrow R^m. - A matrix nxn matrix is Identity
*matrix*, corresponds to zero transformation from .

**Example**

Let’s consider the linear transformation from R^{2} \rightarrow R^3 such that:

Now, we will be verifying that it is a linear transformation. For that we need to check for the above two conditions for the Linear mapping, first, we will be checking the constant multiplicative conditions:

and the following transformation:

It proves that the above transformation is Linear transformation. Examples of not linear transformation include trigonometric transformation, polynomial transformations.

### Kernel/ Range Space:

#### Kernel space:

Let T: V \rightarrow W is linear transformation then \forall v \epsilon V such that:

is the kernel space of T. It is also known as null space of T.

- The kernel space of zero transformation for T:V \rightarrow W is W.
- The kernel space of identity transformation for T:V \rightarrow W is {0}.

The dimensions of the kernel space is known as nullity or null(T).

#### Range Space:

Let T: V \rightarrow W is linear transformation then \forall v \epsilon V such that:

is the range space of T. Range space is always non-empty set for a linear transformation on matrix because:

The dimensions of the range space is known as rank (T). The sum of rank and nullity is the dimension of the domain:

### Linear Transformation as Rotation

Some of the transformation operators when applied to some vector give the output of vector with rotation with angle \theta of the original vector.

- The linear transformation T: R^2 \rightarrow R^2 given by matrix: has the property that it rotates every vector in anti-clockwise about the origin wrt angle \theta:

Let v

which is similar to rotating the original vector by \theta.

### Linear Transformation as Projection

A linear transformation T: R^3 \rightarrow R^3 is given by:

T =

If a vector is given by v = (x, y, z) . Then, T\cdot v = (x, y, 0). That is the orthogonal projection of original vector.

### Differentiation as Linear Transformation

Let T: be the differentiation transformation such that: Then for two polynomials p(z), , we have:

Similarly, for the scalar a \epsilon F we have:

The above equation proved that differentiation is linear transformation.

### References:

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