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The composition of two linear transformations $T_1: U \rightarrow V$ and $T_2: V \rightarrow W$ is a linear transformation $T_2 \circ T_1: U \rightarrow W$ defined by $(T_2 \circ T_1)(u) = T_2(T_1(u))$, for all $u \in U$. Example: If $T_1(\vec{x}) = A\vec{x}$ and $T_2(\vec{y}) = B\vec{y}$, then $(T_2 \circ T_1)(\vec{x}) = BA\vec{x}$.

For a linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$, its matrix representation $[T]$ is an $m \times n$ matrix such that $T(\vec{x}) = [T]\vec{x}$ for all $\vec{x} \in \mathbb{R}^n$. Example: A rotation in $\mathbb{R}^2$ can be represented by $[T] = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$.

A reflection transformation is a linear transformation that flips a vector across a subspace. Example: Reflection in $\mathbb{R}^2$ across the x-axis is given by $T(\vec{x}) = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \vec{x}$.

A linear transformation $T: V \rightarrow W$ is an isomorphism if T is both injective (one-to-one) and surjective (onto), hence invertible. Example: $\mathbb{R}^n$ to $\mathbb{R}^n$, where $T(\vec{x}) = A\vec{x}$ and $A$ is an invertible matrix.

The rank-nullity theorem states that for a linear transformation $T: V \rightarrow W$, the dimension of V is the sum of the rank of T and the nullity of T (dimension of the kernel): $\text{dim}(V) = \text{rank}(T) + \text{nullity}(T)$. Example: For a matrix $A \in \mathbb{R}^{n \times m}$, $\text{dim}({\mathbb{R}^m}) = \text{rank}(A) + \text{nullity}(A)$.

The image of a linear transformation $T: V \rightarrow W$ is the set of all vectors in W that are $T(v)$ for some $v \in V$. Example: For linear map $T(\vec{x}) = A\vec{x}$, the image is the span of the columns of $A$.

The determinant of a square matrix $A$, representing a linear transformation $T: V \rightarrow V$, is a scalar value that provides information about the linear transformation, including whether it is invertible (non-zero determinant). Example: For matrix $A$, if $\det(A) \neq 0$, then T is invertible.

A homothety, or dilation, is a linear transformation that enlarges or diminishes the size of an object without altering its shape. Example: Scaling by a factor $k$ in $\mathbb{R}^2$ can be represented by $T(\vec{x}) = k\vec{x}$ or in matrix form by $A = kI$.

A mapping $T: V \rightarrow W$ between two vector spaces V and W over the same field is called a linear transformation if for all vectors $u, v \in V$ and any scalar $c$, the two properties hold: $T(u + v) = T(u) + T(v)$ and $T(cu) = cT(u)$. Example: $T(\vec{x}) = A\vec{x}$ where $A$ is a matrix.

The trace of a square matrix $A$, representing a linear transformation $T: V \rightarrow V$, is the sum of the diagonal entries of $A$. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, the trace is $1 + 4 = 5$.

For a linear transformation $T: V \rightarrow V$, an eigenvalue $\lambda$ is a scalar such that there exists a non-zero vector $\vec{v} \in V$ with $T(\vec{v}) = \lambda \vec{v}$. Example: For matrix $A$, if $A\vec{v} = \lambda \vec{v}$, then $\lambda$ is an eigenvalue of $A$.

A linear transformation $T: V \rightarrow V$ is a projection if $T^2 = T$. It projects vectors onto a subspace of V. Example: Projection onto a line in $\mathbb{R}^2$ given by $T(\vec{x}) = \frac{\vec{x} \cdot \vec{a}}{\vec{a} \cdot \vec{a}} \vec{a}$, where $\vec{a}$ is a vector on the line.

A rotation transformation is a linear transformation that rotates vectors in a plane about the origin or a fixed point. Example: In $\mathbb{R}^2$, rotation by $\theta$ is given by $T(\vec{x}) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \vec{x}$.

The kernel of a linear transformation $T: V \rightarrow W$ is the set of all vectors $v \in V$ such that $T(v) = \vec{0}$. Example: For $T(\vec{x}) = A\vec{x}$, the kernel is all $\vec{x}$ that satisfy $A\vec{x} = \vec{0}$.

An eigenvector of a linear transformation $T: V \rightarrow V$ is a non-zero vector $\vec{v} \in V$ that satisfies $T(\vec{v}) = \lambda \vec{v}$ for some scalar $\lambda$, which is the associated eigenvalue. Example: For matrix $A$, if $A\vec{v} = \lambda \vec{v}$, then $\vec{v}$ is an eigenvector of $A$.

An eigenbasis of a linear transformation $T: V \rightarrow V$ is a basis for V consisting of eigenvectors of T. Example: If matrix $A$ has $n$ linearly independent eigenvectors, they form an eigenbasis for $\mathbb{R}^n$.

A linear transformation $T: V \rightarrow W$ has an inverse $T^{-1}: W \rightarrow V$ if there exists a map $T^{-1}$ such that $T^{-1}(T(v)) = v$ for all $v \in V$ and $T(T^{-1}(w)) = w$ for all $w \in W$. Example: If $T(\vec{x}) = A\vec{x}$ is invertible, then $T^{-1}(\vec{y}) = A^{-1}\vec{y}$.

A linear functional is a linear transformation from a vector space V to its field of scalars. Example: For $\mathbb{R}^n$, any linear map that takes a vector $\vec{x}$ to a single real number (like a dot product with a fixed vector).

A linear transformation $T: V \rightarrow V$ is diagonalizable if there exists a basis for V consisting of eigenvectors of T, making the matrix representation of T a diagonal matrix. Example: If matrix $A$ has $n$ independent eigenvectors, it is diagonalizable.

A matrix $A$ is an orthogonal matrix if $A^TA = I$, which means its rows and columns are orthogonal unit vectors. It represents a linear transformation that preserves lengths and angles. Example: Rotation matrices in $\mathbb{R}^2$ and $\mathbb{R}^3$.