powerpoint presentation on linear transformation

Linear Transformations

6. linear transformations #.

You will already be familiar with the use of functions in mathematics to study sets and the mapping from a set of inputs set to another set of outputs. The notation used to denote functions is of the form \(f: X \to Y\) where \(f\) is the name of the function, \(X\) is the set of inputs known as the domain and \(Y\) is a member of the set of outputs known as the codomain . The mapping which defines the relationship between the domain and codmain is defined using \(y = f(x)\) where \(x\) is a member of the domain and \(y\) is a member of the codomain.

In linear algebra we study the linear mapping of a set of vectors to another set of vectors so we define \(T: V \to W\) where \(V\) and \(W\) are vector spaces and the mapping from an input vector \(\mathbf{u} \in V\) to an output vector \(\mathbf{w} \in W\) is given by \(\mathbf{w} = T(\mathbf{u})\) .

Linear transformations have lots of uses in mathematics and computing. A good example is in the field of computer graphics and computer games where they are fundamental to the manipulation and visualisation of three-dimensional objects.

We begin with the formal definition of a linear transformation.

Definition 6.1 (Linear transformation)

If \(V\) and \(W\) are two vector spaces over the same field \(F\) then by a linear transformation (or linear mapping ) is a mapping \(T: V \to W\) that for any two vectors \(\mathbf{u}, \mathbf{v} \in V\) and any scalar \(\alpha \in F\) the following conditions hold

addition operation: \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\) ;

scalar multiplication: \(T(\alpha \mathbf{u}) = \alpha T(\mathbf{u})\) .

The result of applying a linear transformation to an object is known as the image .

For example, let \(V = \mathbb{R}^2\) and \(W = \mathbb{R}^3\) then \(T : V \to W\) defined by \(T : (x, y)^\mathsf{T} \mapsto (x, y, 0)^\mathsf{T}\) is a linear transformation. Let \(\mathbf{u} = (u_1, u_2)^\mathsf{T}, \mathbf{v} = (v_1, v_2)^\mathsf{T} \in \mathbb{R}^2\) and \(\alpha \in \mathbb{R}\) then

so \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\) and the addition condition is satisfied. Similarly

so \(T(\alpha \mathbf{u}) = \alpha T(\mathbf{u})\) and the scalar multiplication condition is satisfied and combined with the addition condition we have shown that \(T\) is a linear transformation. We can combine the addition and scalar multiplication conditions to give a single condition.

Definition 6.2 (Linear transformation condition)

A transformation \(T : V \to W\) is a linear transformation if the following condition is satisfied for any \(\mathbf{u}, \mathbf{v} \in V\) and \(\alpha \in F\)

Example 6.1

Determine which of the following transformations are linear transformations

(i)   \(T: \mathbb{R}^3 \to \mathbb{R}^2\) defined by \(T: (x, y, z)^\mathsf{T} \mapsto (x, y)^\mathsf{T}\)

Let \(\mathbf{u} = (u_1, u_2, u_3)^\mathsf{T}, \mathbf{v} = (v_1, v_2, v_3)^\mathsf{T} \in \mathbb{R}^3\) and \(\alpha \in \mathbb{R}\) then

Since \(T(\mathbf{u} + \alpha \mathbf{v}) = T(\mathbf{u}) + \alpha T(\mathbf{v})\) then \(T: (x, y, z)^\mathsf{T} \mapsto (x, y)^\mathsf{T}\) is a linear transformation.

(ii)   \(T: \mathbb{R}^3 \to \mathbb{R}^2\) defined by \(T: (x, y, z)^\mathsf{T} \mapsto (x + 3, y)^\mathsf{T}\)

Since \(T(\mathbf{u} + \alpha \mathbf{v}) \neq T(\mathbf{u}) + \alpha T(\mathbf{v})\) then \(T: (x, y, z)^\mathsf{T} \mapsto (x + 3, y)^\mathsf{T}\) is not a linear transformation.

Note that we could have shown this by a counterexample, e.g., let \(\mathbf{u} = ( 1, 0 , 0 )^\mathsf{T}, \mathbf{v} = (2, 0, 0)^\mathsf{T} \in \mathbb{R}^3\) then

(iii)   \(T: P(\mathbb{R}) \to P(\mathbb{R})\) defined by \(T: p \mapsto p \dfrac{\mathrm{d}p}{\mathrm{d}x}\)

Let \(u = x \in P(\mathbb{R})\) then

therefore \(T(2u) \neq 2T(u)\) and \(T: p \mapsto p \dfrac{\mathrm{d}p}{\mathrm{d}x}\) is not a linear transformation.

6.1. Transformation matrices #

For convenience we tend to use matrices to represent linear transformations. Let \(T: V \to W\) be a linear transformation from the vector spaces \(V\) to \(W\) where \(V, W \in \mathbb{R}^n\) . If \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\) is a basis for \(V\) then for a vector \(\mathbf{u} \in V\)

and by the definition of a linear transformation we can apply a linear transformation \(T\) to the vectors \(\mathbf{u}\) and \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\)

so \(T(\mathbf{u})\) depends on the vectors \(T(\mathbf{v}_1), T(\mathbf{v}_2), \ldots, T(\mathbf{v}_n)\) . We can write this as the matrix equation

In other words we can apply a linear transformation simply by multiplying \(\mathbf{u}\) by a matrix \(A\) .

Definition 6.3 (Transformation matrix)

Let \(T : V \to W\) be a linear transformation and \(A\) be a matrix such that

\(A\) is said to be the matrix representation of the linear transformation \(T\) (also known as the transformation matrix ).

Example 6.2

A linear transformation \(T:\mathbb{R}^2 \to \mathbb{R}^2\) is defined by \(T: (x, y)^\mathsf{T} \mapsto (3x + y, x + 2y)^\mathsf{T}\) . Calculate the transformation matrix and use it to calculate \(T(1,1)^\mathsf{T}\) .

Since we are mapping from \(\mathbb{R}^2\) the transformation matrix is

Applying the transformation to the standard basis vectors

so the transformation matrix is

Applying the transformation matrix to \((1, 1)^\mathsf{T}\)

The affects of the linear transformation from Example 6.1 is illustrated in Fig. 6.1 . Note that the transformation \(T\) can be thought of as changing the basis of the vector space. The unit square with respect to the basis \(\{\mathbf{e}_1, \mathbf{e}_1\}\) has been transformed into a unit parallelogram with respect to the basis \(\{ T(\mathbf{e}_1), T(\mathbf{e}_2)\}\) .

Fig. 6.1 The affect of applying a linear transformation \(T: (x,y)^\mathsf{T} \mapsto (3x + y, x + 2y)^\mathsf{T}\) to the vector \((1,1)^\mathsf{T}\) . #

6.2. Finding the transformation matrix from a set of images #

The calculation of the transformation matrix in Example 6.2 was straightforward as we knew what the transformation was. This will not always be a the case and we may know what the output of the transformation (known as the image) is but not the transformation itself. Consider a linear transformation \(T: \mathbb{R}^n \to \mathbb{R}^m\) applied to vectors \(\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n\) . If \(A\) is the transformation matrix for \(T\) then

Theorem 6.1 (Determining the linear transformation given the inputs and image vectors)

Given a linear transformation \(T: \mathbb{R}^n \to \mathbb{R}^m\) applied to a set of \(n\) vectors \(\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n\) with known image vectors \(T(\mathbf{u}_1), T(\mathbf{u}_2), \ldots, T(\mathbf{u}_n)\) then the transformation matrix \(A\) for \(T\) is

Example 6.3

Determine the transformation matrix \(A\) for the linear transformation \(T\) such that

The inverse of \((\mathbf{u}_1, \mathbf{u}_2)\) is

Right multiplying the image matrix

This is the transformation matrix from Example 6.2 .

6.3. Inverse linear transformation #

Definition 6.4 (Inverse linear transformation)

Let \(T: V \to W\) be a linear transformation with the transformation matrix \(A\) then \(T\) has an inverse transformation denoted by \(T^{-1}: W \to V\) which reverses the affects of \(T\) . If \(\mathbf{u} \in V\) and \(\mathbf{v} \in W\) then

where \(A^{-1}\) is the transformation matrix for \(T^{-1}\) .

Example 6.4

Determine the inverse of the transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\) defined by \(T(x, y)^\mathsf{T} \mapsto (3 x + y, x + 2 y)^\mathsf{T}\) and calculate \(T^{-1}(4,3)^\mathsf{T}\) .

We saw in Example 6.2 that the transformation matrix for \(T\) is

which has the inverse

Determining the inverse transformation

Calculating \(T^{-1}\begin{pmatrix} 4 \\ 3 \end{pmatrix}\)

• Preferences

Chapter 4 Linear Transformations - PowerPoint PPT Presentation

Chapter 4 Linear Transformations

A linear transformation is a function t that maps a vector space v into another ... the preimage of w: the set of all v in v such that t(v)=w. 6 - 3. notes: ... – powerpoint ppt presentation.

• 4.1 Introduction to Linear Transformations
• 4.2 The Kernel and Range of a Linear Transformation
• 4.3 Matrices for Linear Transformations
• 4.4 Transition Matrices and Similarity
• A linear transformation is a function T that maps a vector space V into another vector space W
• Image of v under T
• the range of T
• The set of all images of vectors in V.
• the preimage of w
• The set of all v in V such that T(v)w.
• Ex (Verifying a linear transformation T from R2 into R2)
• Ex (Functions that are not linear transformations)
• Notes Two uses of the term linear.
• Zero transformation
• Identity transformation
• Thm 4.1 (Properties of linear transformations)
• Ex (Linear transformations and bases)
• Ex (A linear transformation defined by a matrix)
• Thm 4.2 (The linear transformation given by a matrix)
• Rotation in the plane
• A projection in R3
• A linear transformation from Mm?n into Mn ?m
• Kernel of a linear transformation T
• Ex 2 The kernel of the zero and identity transformations
• Finding the kernel of a linear transformation
• Thm 4.3 (The kernel is a subspace of V)
• The kernel of T is also called the nullspace of T.
• Corollary to Thm 4.3
• Thm 4.4 The range of T is a subspace of W
• Rank of a linear transformation T V?W
• Thm 4.5 Sum of rank and nullity
• Finding the rank and nullity of a linear transformation
• Thm 4.6 (One-to-one linear transformation)
• One-to-one and not one-to-one linear transformation
• Thm 4.7 (Onto linear transformation)
• Thm 4.8 (One-to-one and onto linear transformation)
• Isomorphism
• Ex (Isomorphic vector spaces)
• Two representations of the linear transformation TR3?R3
• Three reasons for matrix representation of a linear transformation
• It is simpler to write.
• It is simpler to read.
• It is more easily adapted for computer use.
• Thm 4.10 (Standard matrix for a linear transformation)
• Ex (Finding the standard matrix of a linear transformation)
• Composition of T1 Rn?Rm with T2 Rm?Rp
• Ex (The standard matrix of a composition)
• Inverse linear transformation
• If the transformation T is invertible, then the inverse is unique and denoted by T1 .
• Existence of an inverse transformation
• T is invertible.
• T is an isomorphism.
• A is invertible.
• If T is invertible with standard matrix A, then the standard matrix for T1 is A1 .
• Ex (Finding the inverse of a linear transformation)
• the matrix of T relative to the bases B and B'
• Transformation matrix for nonstandard bases
• Ex (Finding a transformation matrix relative to nonstandard bases)
• Two ways to get from to
• Similar matrix
• For square matrices A and A of order n, A is said to be similar to A if there exist an invertible matrix P such that
• Ex (A comparison of two matrices for a linear transformation)

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Linear Transformations

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Abstract: Studying quantum correlations in the presence of loss is of critical importance for the physical modeling of real quantum systems. Here, we demonstrate the control of spatial correlations between entangled photons in a photonic chip, designed and modeled using the singular value decomposition approach. We show that engineered loss, using an auxiliary waveguide, allows one to invert the spatial statistics from bunching to antibunching. Furthermore, we study the photon statistics within the loss-emulating channel and observe photon coincidences, which may provide insights into the design of quantum photonic integrated chips.

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