Math 511: Linear Algebra
6.5 Applications
6.5.1 Isometries¶
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Many elementary linear transformations perform geometrical transformations on a vector space $V$.
Definition¶
A linear transformation from whose domain and codomain are the same vector space $L:V\rightarrow V$ is said to act on the vector space because the input vector from vector space $V$ is transformed to an output vector in vector space $V$. A matrix that represents a linear transformation acting on a finite dimensional vector space like $\mathbb{R}^n$ is a square matrix.
Section 5.3.7 introduces the properties or orthogonal matrices. Orthogonal matrices represent linear transformations on a finite dimensional inner product space that preserves both magnitude and angle. That is, matrix $A$ is orthogonal if and only if
- the magnitude of a vector input is also the magnitude of the vector output, $\left\|A\mathbf{x}\right\| = \left\|\mathbf{x}\right\|$
- the angle between to vector inputs is also the angle between their outputs, $\langle A\mathbf{x},A\mathbf{y}\rangle = \langle \mathbf{x},\mathbf{y}\rangle$.
Definition¶
A linear transformation on an inner product space $V$, $L\,:\,V\rightarrow V$, that preserves the distances between two vectors is called an isometry. Thus if $L(\mathbf{x})=A\mathbf{x}$ and $\mathbf{x},\ \mathbf{y}\in V$ are vectors in inner product space $V$, then
$$ \left\|A\mathbf{y}-A\mathbf{x}\right\| = \left\|A\left(\mathbf{y}-\mathbf{x}\right)\right\| = \left\|\mathbf{y}-\mathbf{x}\right\| $$
Since vector $\mathbf{x}\in V$ can be the zero vector in this definition, isometries also preserve the magnitudes of vectors. However the angle between vectors may not be preserved.
Isometries are also called rigid transformations because the image of a geometric figure in an inner product space will be a congruent geometrical figure. The image may lie in a new position in the inner product space, or sit in a new orientation, however its size and shape is unchanged. Isometries preserve a geometric figure's size and shape.
Isometries are called rigid motion because geometrical figures in the domain can be flipped, rotated, slid, moved, or translated into a new position in the inner product space without changing its size or shape.
For example a reflection mirrors geometrical shapes in a finite dimensional vector space to another position and orientation in the vector space
6.5.2 Affine Subspaces¶
Affine subspaces are not really subspaces of a vector space because they are not closed under linear combinations. Instead, they are subspaces of a vector space that has been translated away from the origin.
Definition¶
An affine subspace $B$ in vector space $V$ is a subset of $V$ in which the set $S = \left\{ \mathbf{b}-\mathbf{a}\,:\,\mathbf{b}\in B\right\}$ is a subspace of $V$ for some fixed vector $\mathbf{a}\in V$.
Similarly, one can start with a subspace $S$ of vector space $V$ and a nonzero vector $\mathbf{a}\in V$. Then create the new set
$$ \mathbf{a}+S = \left\{ \mathbf{a}+\mathbf{x}\,:\,\mathbf{x}\in S\right\} $$
Affine spaces have been discussed in these notes before. A translation is a transformation on a vector space that models movement from one location to another. A translation is a rigid transformation, however it is not a linear transformation because the origin does not stay fixed. The zero vector gets moved to a new location with all of the other vectors.
Definition¶
A translation is an isometry on a vector space $V$ that adds a fixed vector $\mathbf{a}\in V$ to every vector in $V$.
$$ L(\mathbf{x}) := \mathbf{x} + \mathbf{a} $$
6.5.3 More Geometry¶
We need a few more definitions before we can talk about simple geomety of some linear transformations.
A hyperplane is a generalization of a line in $\mathbb{R}^2$ and a plane in $\mathbb{R}^3$.
Definition¶
A hyperplane in a n-dimensional vector space is an (n-1)-dimensional subspace, or (n-1) dimensional affine subspace.
Like a line in $\mathbb{R}^2$ and a plane in $\mathbb{R}^3$, a hyperplane separates a vector space into to equal (if infinite) halves.
In an inner product space, a hyper plane may be described as the set of all vectors orthogonal to a fixed normal vector. For a fixed vector $\mathbf{n}\in V$, the hyperplane $H$ is defined by
$$ H = \left\{ \mathbf{x}\in V\,:\, \langle \mathbf{n},\mathbf{x} \rangle = 0 \right\} $$
Recall that for any linear transformation from vector space $V$ to vector space $W$, $L\,:\,V\rightarrow W$, the zero vector in $V$ must be mapped to the zero vector in $W$
$$ L\left(\mathbf{0}_V\right) = \mathbf{0}_W $$
For a linear transformation on vector space $V$ we have that $L\left(\mathbf{0}_V\right) = \mathbf{0}_V$. Whenever a vector is mapped by a transformation to itself, we say that the vector is a fixed point of the transformation.
Definition¶
If $L$ is a transformation on vector space $V$, then any vector $\mathbf{x}\in V$ so that
$$ L(\mathbf{x}) = \mathbf{x} $$
is called a fixed point of $L$.
So for a linear transformation on vector space $V$, the zero vector is always a fixed point.
Translations have no fixed points, so they cannot be a linear transformation.
6.5.4 Reflections¶
A reflection on inner product space $\mathbb{R}^2$ maps each vector in $\mathbb{R}^2$ to its mirror image with respect to a line in $\mathbf{R}^2$. This line can be a subspace of $\mathbb{R}^2$, or an affine subspace of $\mathbb{R}^2$.
Example 1¶
Reflection about the $y$-axis¶
The reflection $R\,:\,\mathbb{R}^2\rightarrow\mathbb{R}^2$ given by
$$ R(\mathbf{x}) = A\mathbf{x} = \begin{bmatrix} -1\ &\ \ 0\ \\ \ \ 0\ &\ \ 1\ \end{bmatrix}\mathbf{x} $$
maps every vector $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ to $\begin{bmatrix} -x_1 \\ \ \ x_2 \end{bmatrix}$.
Example 2¶
Reflection across the $x$-axis¶
The reflection $T\,:\,\mathbb{R}^2\rightarrow\mathbb{R}^2$ defined by
$$ T(\mathbf{x}) = B\mathbf{x} = \begin{bmatrix}\ \ 1\ &\ \ 0\ \\ \ \ 0\ & -1\ \end{bmatrix}\mathbf{x} $$
maps every vector $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ to $\begin{bmatrix}\ \ x_1 \\ -x_2 \end{bmatrix}$.
Example 3¶
Reflection across the plane $y=x$ in $\mathbb{R}^3$¶
The reflection $L\,:\,\mathbb{R}^3\rightarrow\mathbb{R}^3$ defined by
$$ L(\mathbf{x}) = A\mathbf{x} = \begin{bmatrix}\ 0\ &\ 1\ &\ 0\ \\ \ 1\ &\ 0\ &\ 0\ \\ \ 0\ &\ 0\ &\ 1\ \end{bmatrix}\mathbf{x} $$
maps every vector $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$ to $\begin{bmatrix} x_2 \\ x_1 \\ x_3 \end{bmatrix}$.
Definition¶
A reflection on a finite dimensional inner product space $V$ is an isometry on $V$ with a hyperplane of fixed points.
The hyperplane of fixed points is the mirror for vectors not on the hyperplane. A reflection is a rigid transformation because it is an isometry. The orthogonal distance from each vector $\mathbf{x}\in V$ from the hyperplane is also the distance of the image vector $L(\mathbf{x})$ from the hyperplane, on the other side of the hyperplane.
The hyperplane of fixed points is called the axis of symmetry for the reflection.
6.5.5 Dilations and Contractions¶
Dilations and Contractions stretch or squeeze vectors along a line or linear subspace. A linear transformation can stretch vectors in one direction while squeezing them in another direction.
Example 4¶
Dilation along the $x$-axis in $\mathbb{R}^2$¶
$$ L(\mathbf{x}) = A\mathbf{x} = \begin{bmatrix}\ 2\ &\ 0\ \\ \ 0\ &\ 1\ \end{bmatrix} $$
Clearly a contraction or dilation is not a rigid transformation.
6.5.6 Shear¶
A shear occurs in mechanics and fluid dynamics. Shear winds can make landing an airplane difficult or impossible.
Example 5¶
A shear parallel to the $x$-axis¶
$$ L(\mathbf{x}) = A\mathbf{x} = \begin{bmatrix}\ 1\ &\ 2\ \\ \ 0\ &\ 1\ \end{bmatrix} $$
The linear transformation adds a vector twice the distance of vector $\langle x,y \rangle$ from the $x$-axis and with direction parallel to the $x$-axis.
- The distance of the vector $\langle x,y \rangle$ from the $x$-axis is $y$
- The direction parallel to the $x$-axis is the vector $\ihat$
So the shear adds $2y\ihat = \langle 2y, 0 \rangle$ to vector $\langle x, y \rangle$,
$$ L\left(\langle x, y \rangle\right) = \langle x+2y, y \rangle $$.
We can construct the standard matrix representation of the shear $L$ by determining the images of $\ihat$ and $\jhat$.
$$ \begin{align*} L(\ihat) &= \begin{bmatrix} 1 + 2\cdot 0 \\ 0 \end{bmatrix} = \ihat \\ \\ L(\jhat) &= \begin{bmatrix} 0 + 2\cdot 1 \\ 1 \end{bmatrix} = \begin{bmatrix}\ 2\ \\ \ 1\ \end{bmatrix} \end{align*} $$
Definition¶
A shear on a finite dimensional inner product space $V$ is an an linear transformation that translates each vector $\mathbf{x}\in V$ a length proportional to the distance of vector $\mathbf{x}$ to a fixed hyperplane and parallel to that hyperplane. A shear is also called a transvection or shear transformation.
In $\mathbb{R}^2$, a shear translates a vector a distance proportional to its distance from a line, and in a direction parallel to the line.
In $\mathbb{R}^3$, a shear translates a vector a distance proportional to its distance from a plane, and in a direction parallel to the plane.
A transvection (or shear) is not a rigid transformation because the amount of translation is proportional to its distance from the hyperplane. Geometric shapes will be dilated or contracted parallel to the hyperplane, but not perpendicular to the hyperplane.
(i) In $\mathbb{R}^2$, the images of squares are parallelograms, and the images of circles are ellipses.
(ii) In $\mathbb{R}^3$, the images of cubes are parallelepipeds and the images of spheres are ellipsoids.
A shear matrix is a type III elementary matrix, or type III elementary column matrix.
6.5.7 Rotations¶
Rotations are easy to explain in $\mathbb{R}^2$, complicated to explain in $\mathbb{R}^3$, difficult to explain in $\mathbb{R}^4$, and it just keeps getting increasingly convoluted. Rotations in abstract vector spaces can be hard.
Fortunately, we can explain rotations in terms of their matrix represenations.
Definition¶
An $n\times n$ orthogonal matrix whose determinant equals $1$ represents a rotation on $\mathbb{R}^n$.
Recall that the determinant of an orthogonal matrix is always $\pm 1$. If $Q\in\mathbb{R}^{n\times n}$ is an orthogonal matrix, then
$$ 1 = \det(I_n) = \det\left(QQ^{-1}\right) = \det\left(QQ^T\right) = \det(Q)\det(Q^T) = \det(Q)\det(Q) = \det(Q)^2 $$
Our definition says that the matrix representation of a convoluted rotation in $n$-dimensional space will be an orthogonal matrix with positive determinant.
In general, orthogonal matrices represent rotations, reflections or both. Clearly if an orthogonal matrix includes a reflection, then the determinant will be negative. However one cannot tell from a negative determinant whether the linear transformation also includes a rotation.
The only fixed point of a rotation is the origin. The zero vector doesn't really have a direction so a rotation can't move it to another vector.
Recall that reflections have an entire hyperplane of fixed points. So if your orthogonal matrix has a fixed point that is not the zero vector, then it has determinant -1 and there is no rotation.
There is a caveat to this. Ask your professor about reflections across the origin. This is not strictly a reflection in a vector space with dimension greater than one according to our definition, but it is still called a reflection.
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