Linear Algebra 101 Unit 2 ReviewLinear Transformations

What Are Linear Transformations?

• Linear transformations map vectors from one vector space to another while preserving vector addition and scalar multiplication
• Denoted as $T: V \rightarrow W$, where $V$ and $W$ are vector spaces and $T$ is the linear transformation
• For any vectors $\vec{u}, \vec{v} \in V$ and scalar $c$, a linear transformation satisfies:
  • Additivity: $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$
  • Homogeneity: $T(c\vec{u}) = cT(\vec{u})$
• Linear transformations can be represented using matrices, with the matrix acting on the input vector to produce the output vector
• Examples of linear transformations include rotations, reflections, and projections in 2D or 3D space
• Linear transformations preserve the origin, meaning $T(\vec{0}) = \vec{0}$
• Geometrically, linear transformations maintain the relative positions of vectors and the straightness of lines

Key Properties of Linear Transformations

• Linearity is the defining property of linear transformations, consisting of additivity and homogeneity
• Linear transformations are uniquely determined by their action on basis vectors
  • If $\{\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n\}$ is a basis for $V$, then $T$ is completely determined by $T(\vec{v}_1), T(\vec{v}_2), \ldots, T(\vec{v}_n)$
• The kernel (or null space) of a linear transformation $T$ is the set of all vectors $\vec{v} \in V$ such that $T(\vec{v}) = \vec{0}$
  • Denoted as $\ker(T)$ or $\text{null}(T)$
• The range (or image) of a linear transformation $T$ is the set of all vectors $\vec{w} \in W$ such that $\vec{w} = T(\vec{v})$ for some $\vec{v} \in V$
  • Denoted as $\text{range}(T)$ or $\text{im}(T)$
• The rank of a linear transformation is the dimension of its range
• The nullity of a linear transformation is the dimension of its kernel
• The rank-nullity theorem states that for a linear transformation $T: V \rightarrow W$, $\dim(V) = \text{rank}(T) + \text{nullity}(T)$

Matrix Representation of Linear Transformations

• Linear transformations can be represented using matrices, with the matrix acting on the input vector to produce the output vector
• If $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a linear transformation and $\{\vec{e}_1, \vec{e}_2, \ldots, \vec{e}_n\}$ is the standard basis for $\mathbb{R}^n$, then the matrix representation of $T$ is:
  • $A = [T(\vec{e}_1) \quad T(\vec{e}_2) \quad \ldots \quad T(\vec{e}_n)]$
• The columns of the matrix $A$ are the images of the basis vectors under the linear transformation $T$
• For any vector $\vec{v} \in \mathbb{R}^n$, $T(\vec{v}) = A\vec{v}$, where $A\vec{v}$ represents matrix-vector multiplication
• The matrix representation depends on the choice of bases for the domain and codomain vector spaces
• Changing the bases results in a different matrix representation for the same linear transformation
• The matrix representation allows for efficient computation and analysis of linear transformations using matrix algebra

Common Types of Linear Transformations

• Rotation: A linear transformation that rotates vectors by a specified angle around the origin
  • In 2D, a rotation by angle $\theta$ is represented by the matrix $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$
• Reflection: A linear transformation that reflects vectors across a line or plane passing through the origin
  • In 2D, a reflection across the x-axis is represented by the matrix $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
• Projection: A linear transformation that projects vectors onto a specified subspace
  • In 2D, a projection onto the x-axis is represented by the matrix $\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$
• Scaling: A linear transformation that stretches or compresses vectors by a specified factor along each coordinate axis
  • In 2D, scaling by factors $a$ and $b$ is represented by the matrix $\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$
• Shear: A linear transformation that shifts vectors parallel to a coordinate axis by an amount proportional to their coordinate along another axis
  • In 2D, a horizontal shear by factor $k$ is represented by the matrix $\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}$
• Identity transformation: A linear transformation that maps every vector to itself, represented by the identity matrix

Composition and Inverse of Linear Transformations

• Composition of linear transformations is the process of applying one linear transformation followed by another
  • If $T: V \rightarrow W$ and $S: W \rightarrow U$ are linear transformations, their composition $S \circ T: V \rightarrow U$ is defined as $(S \circ T)(\vec{v}) = S(T(\vec{v}))$ for all $\vec{v} \in V$
• The composition of linear transformations is associative: $(T \circ S) \circ R = T \circ (S \circ R)$
• The matrix representation of the composition of linear transformations is the product of their individual matrix representations
  • If $A$ and $B$ are the matrix representations of $T$ and $S$, respectively, then the matrix representation of $S \circ T$ is $BA$
• The inverse of a linear transformation $T: V \rightarrow V$ is a linear transformation $T^{-1}: V \rightarrow V$ such that $T \circ T^{-1} = T^{-1} \circ T = I$, where $I$ is the identity transformation
• A linear transformation is invertible (or nonsingular) if and only if it is bijective (one-to-one and onto)
• The matrix representation of the inverse of a linear transformation is the inverse of its matrix representation
  • If $A$ is the matrix representation of $T$, then $A^{-1}$ is the matrix representation of $T^{-1}$, provided $A$ is invertible

Eigenvalues and Eigenvectors in Linear Transformations

• An eigenvector of a linear transformation $T: V \rightarrow V$ is a nonzero vector $\vec{v} \in V$ such that $T(\vec{v}) = \lambda\vec{v}$ for some scalar $\lambda$
  • The scalar $\lambda$ is called the eigenvalue corresponding to the eigenvector $\vec{v}$
• Eigenvectors are vectors that, when acted upon by a linear transformation, only change by a scalar factor (the eigenvalue)
• To find the eigenvalues of a linear transformation, solve the characteristic equation $\det(A - \lambda I) = 0$, where $A$ is the matrix representation of $T$ and $I$ is the identity matrix
• For each eigenvalue $\lambda$, find the corresponding eigenvectors by solving the equation $(A - \lambda I)\vec{v} = \vec{0}$
• Eigenvectors corresponding to distinct eigenvalues are linearly independent
• The set of all eigenvectors corresponding to an eigenvalue, along with the zero vector, forms an eigenspace
• Eigenvalues and eigenvectors have numerous applications, such as in matrix diagonalization, systems of differential equations, and principal component analysis

Applications of Linear Transformations

• Computer graphics: Linear transformations are used to manipulate and transform 2D and 3D objects, such as in video games and animations
  • Rotations, reflections, scaling, and shearing are common linear transformations used in computer graphics
• Image processing: Linear transformations are used to apply filters, enhance features, and perform image compression
  • Examples include edge detection, blurring, and color space conversions
• Quantum mechanics: Linear transformations are used to describe the evolution of quantum states and the action of quantum operators
  • Unitary transformations, which preserve inner products, are particularly important in quantum mechanics
• Cryptography: Linear transformations are used in various encryption and decryption algorithms
  • The Hill cipher, for example, uses matrix multiplication to encrypt and decrypt messages
• Machine learning: Linear transformations are used in feature extraction, dimensionality reduction, and data preprocessing
  • Principal component analysis (PCA) and linear discriminant analysis (LDA) are examples of linear transformation techniques used in machine learning
• Robotics: Linear transformations are used to describe the motion and orientation of robotic arms and manipulators
  • Homogeneous coordinates and transformation matrices are commonly used in robotics to represent translations and rotations in 3D space

Practice Problems and Examples

1. Given the linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defined by $T(x, y) = (2x - y, x + y)$, find the matrix representation of $T$ with respect to the standard basis.
2. Determine whether the transformation $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined by $T(x, y, z) = (x + y, y + z, z)$ is linear. If it is linear, find its kernel and range.
3. Given the matrix $A = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$, find the linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ represented by $A$.
4. Consider the linear transformations $S(x, y) = (x - y, x + y)$ and $T(x, y) = (3x, -2y)$. Find the matrix representations of $S$ and $T$, and then find the matrix representation of the composition $T \circ S$.
5. Find the eigenvalues and corresponding eigenvectors of the linear transformation represented by the matrix $A = \begin{bmatrix} 4 & -2 \\ 1 & 3 \end{bmatrix}$.
6. A linear transformation $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ is defined by $T(1, 0, 0) = (2, 1, 1)$, $T(0, 1, 0) = (1, 2, 1)$, and $T(0, 0, 1) = (1, 1, 2)$. Find the matrix representation of $T$ and determine whether $T$ is invertible. If it is invertible, find the matrix representation of $T^{-1}$.