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Linear mapping

  • Last Updated : 08 Oct, 2021
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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 \\epsilon V          and a scalar c \\epsilon          K:

  • If the transformation is additive in nature:

f(u + v) = f(u) + f(v)

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

f(cu) = c \cdot f(u)

Zero/Identity Transformation

A  linear transformation T: V \rightarrow V          from a vector space into itself is called Linear operator:

  • Zero-Transformation: For a transformation T: V \rightarrow W          is called zero-transformation if:

T(v) = 0 \, \forall \, V



  • Identity-Transformation: For a transformation T: V \rightarrow V          is called identity-transformation if:

T(v) =v \, \forall \, V

Properties of Linear Transformation

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

  • T(0) =0
  • T(-v) = - T(v)
  • T(u-v) = T(u) - T(v)
  • If v = c_1 v_1 + c_2 v_2 + ... + c_n v_n          then, T(v) = c_1 T(v_1) + c_2 T(v_2) + ... + c_n T(v_n)

Linear Transformation of Matrix

Let T be a mxn matrix, the transformation T:R^n \rightarrow R^m          is  linear transformation if:

T(v) = Av

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 \mathbb{I_n}         , corresponds to zero transformation from R^n \rightarrow R^m         .

A \cdot R^m  = R^n \\ \begin{bmatrix} a_{11}&  a_{12}&  .&  .&  .& a_{1n} \\ a_{21}&  a_{22}&  .&  .&  .&a_{2n} \\ .&  .&  .&  &  & .\\ .&  .&  &  .&  & .\\ .&  .&  &  &  .& .\\ a_{m1}&  a_{m2}&  .&  .&  .&a_{mn} \end{bmatrix} \cdot \begin{bmatrix} v_1\\ v_2\\ .\\ .\\ .\\ v_n \end{bmatrix} = \begin{bmatrix} a_{11} v_1 + a_{12} v_2 \, .\, \, . a_{1n} v_n \\ .\\ .\\ .\\ .\\ a_{m1} v_1 + a_{m2} v_2 \, .\, \, . a_{mn} v_n \\ \end{bmatrix}

Example

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



L(\begin{bmatrix} v_1\\ v_2 \end{bmatrix})= \begin{bmatrix} v_2\\ v_1 - v_2 \\ v_1 + v_2 \end{bmatrix}

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:

L(c \vec{v}) = c \cdot L(\vec{v})

L(c\begin{bmatrix} v_1\\ v_2 \end{bmatrix})= \begin{bmatrix} c v_1\\ c v_1 - c v_2 \\ c v_1 + c v_2 \end{bmatrix}= c \begin{bmatrix} v_1\\ v_1 - v_2 \\ v_1 + v_2 \end{bmatrix} = c L(\vec{v})

and the following transformation:

L(\vec{v} + \vec{w})= L(\vec{v}) + L(\vec{w})

\vec{v} =\begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \vec{w} =\begin{bmatrix} w_1 \\ w_2 \end{bmatrix} \vec{v} + \vec{w} =\begin{bmatrix} v_1 + w_1\\ v_2 + w_2 \end{bmatrix}

L(\vec{v} + \vec{w}) = \begin{bmatrix} v_1 + w_1\\ (v_1 + w_1) - (v_2 + w_2)\\ (v_1 + w_1) + (v_2 + w_2) \end{bmatrix}=\begin{bmatrix} v_1 + w_1\\ (v_1 + v_2) - (w_1 + w_2)\\ (v_1 + v_2) + (w_1 + w_2) \end{bmatrix} = \begin{bmatrix} v_1\\ (v_1 - v_2)\\ (v_1 + v_2) \end{bmatrix} + \begin{bmatrix} w_1\\ (w_1 - w_2)\\ (w_1 + w_2) \end{bmatrix} = L(\vec{v}) + L(\vec{w})

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:

T \cdot v =0

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:

T \cdot v = v

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

T \cdot  0 =0

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

null(T) + rank(T) = dim(V)=n



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: A= \begin{bmatrix} cos\theta & -sin \theta \\ sin\theta & cos \theta \end{bmatrix}         has the property that it rotates every vector in anti-clockwise about the origin wrt angle \theta:

Let v=\begin{bmatrix} r \,  cos \alpha\\ r \, sin \alpha \end{bmatrix}

T(v) = A \cdot v= \begin{bmatrix} cos\theta & -sin \theta \\ sin\theta & cos \theta \end{bmatrix} \cdot \begin{bmatrix} r \, cos \alpha \\ r \, sin \alpha \end{bmatrix} = \begin{bmatrix} r \cdot(\, cos \theta \, cos \alpha - sin \theta \, sin \alpha) \\ r \cdot (\, sin \theta \, cos \alpha + cos \theta \, sin \alpha) \end{bmatrix} = \begin{bmatrix} r \, cos(\theta + \alpha) \\ r \, sin(\theta + \alpha)   \end{bmatrix}

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 = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{bmatrix}

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: P(F) \rightarrow P(F) be the differentiation transformation such that: T \cdot p(z) = p^{'}(z). Then for two polynomials p(z), q(z) \epsilon P(F), we have:

T(p(z) + q(z)) = (p(z) + q(z))^{'} = p^{'}(z) + q^{'}(z) = T(p(z)) + T(q(z))

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

T(a\cdot p(z)) = (a \cdot p(z))^{'} = a p^{'}(z) = a T(p(z))

The above equation proved that differentiation is linear transformation.

References:

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