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# Linear mapping

• Last Updated : 08 Oct, 2021

### 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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