1.2 Linear Transformations and Matrices

Linear transformations are the mathematical superheroes of vector spaces. They preserve vector addition and scalar multiplication, allowing us to map one space to another while keeping the structure intact. These transformations are the backbone of many applications in physics and engineering.

Matrix representations give us a concrete way to work with linear transformations. By expressing transformations as matrices, we can perform calculations and analyze their properties. This powerful tool connects abstract concepts to practical computations, making linear algebra accessible and applicable.

Linear Transformations

Properties of linear transformations

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• Functions between vector spaces that preserve the vector space structure
  • Let $V$ and $W$ be vector spaces over the same field $F$, and let $T: V \to W$ be a function
  • $T$ is a linear transformation if for all vectors $\vec{u}, \vec{v} \in V$ and scalars $c \in F$:
    • Satisfies additivity property: $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$
    • Satisfies homogeneity property: $T(c\vec{u}) = cT(\vec{u})$
• Preserve vector addition and scalar multiplication operations
• Map the zero vector of $V$ to the zero vector of $W$: $T(\vec{0}_V) = \vec{0}_W$
• The set of all linear transformations from $V$ to $W$ is denoted by $\mathcal{L}(V, W)$

Matrix representation of transformations

• Can be represented using matrices
  • Let $V$ and $W$ be finite-dimensional vector spaces with ordered bases $B = \{\vec{v}_1, \ldots, \vec{v}_n\}$ and $C = \{\vec{w}_1, \ldots, \vec{w}_m\}$, respectively
  • For a linear transformation $T: V \to W$, the matrix representation of $T$ with respect to the bases $B$ and $C$ is the $m \times n$ matrix $A = [a_{ij}]$, where $a_{ij}$ is the $i$-th coordinate of $T(\vec{v}_j)$ with respect to the basis $C$
• Matrix representation depends on the choice of bases for the domain and codomain
• Matrix multiplication corresponds to the composition of linear transformations
  • If $T: U \to V$ and $S: V \to W$ are linear transformations with matrix representations $A$ and $B$, respectively, then the matrix representation of the composition $S \circ T: U \to W$ is the product $BA$
• Matrix addition corresponds to the pointwise addition of linear transformations
• Scalar multiplication of matrices corresponds to scalar multiplication of linear transformations

Kernel and range of transformations

• The kernel (or null space) of a linear transformation $T: V \to W$ is the set of all vectors in $V$ that are mapped to the zero vector in $W$
  • $\ker(T) = \{\vec{v} \in V : T(\vec{v}) = \vec{0}_W\}$
  • The kernel is a subspace of the domain $V$
• The range (or image) of a linear transformation $T: V \to W$ is the set of all vectors in $W$ that are the output of $T$ for some input vector in $V$
  • $\text{range}(T) = \{T(\vec{v}) : \vec{v} \in V\}$
  • The range is a subspace of the codomain $W$
• The nullity of a linear transformation $T$ is the dimension of its kernel: $\text{nullity}(T) = \dim(\ker(T))$
• The rank of a linear transformation $T$ is the dimension of its range: $\text{rank}(T) = \dim(\text{range}(T))$
• The rank-nullity theorem relates the dimensions of the domain, kernel, and range of a linear transformation
  • For a linear transformation $T: V \to W$, where $V$ is finite-dimensional: $\dim(V) = \text{rank}(T) + \text{nullity}(T)$

Invertibility of linear transformations

• A linear transformation $T: V \to W$ is invertible (or bijective) if it is both injective (one-to-one) and surjective (onto)
  • Injective: For all $\vec{u}, \vec{v} \in V$, if $T(\vec{u}) = T(\vec{v})$, then $\vec{u} = \vec{v}$
  • Surjective: For every $\vec{w} \in W$, there exists a $\vec{v} \in V$ such that $T(\vec{v}) = \vec{w}$
• If $T$ is invertible, there exists a unique linear transformation $T^{-1}: W \to V$ called the inverse of $T$, such that:
  • $T^{-1} \circ T = I_V$ (the identity transformation on $V$)
  • $T \circ T^{-1} = I_W$ (the identity transformation on $W$)
• A linear transformation $T: V \to W$ is invertible if and only if its kernel is trivial (contains only the zero vector)
  • Equivalently, $T$ is invertible if and only if $\text{nullity}(T) = 0$
• For square matrices (linear transformations from a vector space to itself), a matrix is invertible if and only if its determinant is nonzero
• The inverse of a matrix $A$, denoted by $A^{-1}$, can be found using various methods
  • Gaussian elimination
  • Cramer's rule
  • Adjugate matrix formula