9.3 Linear Transformations and Matrices

Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication. They're crucial in linear algebra, allowing us to understand how vectors change under different operations and providing a link between abstract vector spaces and concrete matrices.

Matrices can represent linear transformations, making complex operations easier to visualize and compute. This connection between transformations and matrices is key to solving many problems in linear algebra, from finding eigenvalues to understanding subspaces like kernels and ranges.

Linear transformations and their properties

Definition and key properties

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• A linear transformation is a function $T$ from a vector space $V$ to a vector space $W$ that preserves vector addition and scalar multiplication
• For any vectors $u$ and $v$ in $V$ and any scalar $c$, a linear transformation $T$ satisfies:
  • $T(u + v) = T(u) + T(v)$ (preserves vector addition)
  • $T(cu) = cT(u)$ (preserves scalar multiplication)
• The zero vector in $V$ is always mapped to the zero vector in $W$ under a linear transformation
  • This follows from the properties of linear transformations: $T(0_V) = T(0 \cdot u) = 0 \cdot T(u) = 0_W$
• Linear transformations preserve linear combinations of vectors
  • If $v_1, v_2, \ldots, v_n$ are vectors in $V$ and $c_1, c_2, \ldots, c_n$ are scalars, then $T(c_1v_1 + c_2v_2 + \ldots + c_nv_n) = c_1T(v_1) + c_2T(v_2) + \ldots + c_nT(v_n)$

Composition of linear transformations

• The composition of two linear transformations is also a linear transformation
  • If $T: U \to V$ and $S: V \to W$ are linear transformations, then their composition $S \circ T: U \to W$ defined by $(S \circ T)(u) = S(T(u))$ is also a linear transformation
• The composition of linear transformations is associative: $(T \circ S) \circ R = T \circ (S \circ R)$
• The identity transformation $I_V: V \to V$ defined by $I_V(v) = v$ for all $v$ in $V$ serves as the identity element for composition

Matrix representation of linear transformations

Representing linear transformations using matrices

• A linear transformation $T$ from an $n$-dimensional vector space $V$ to an $m$-dimensional vector space $W$ can be represented by an $m \times n$ matrix $A$
• The columns of the matrix $A$ are the images of the basis vectors of $V$ under the linear transformation $T$
  • If $\{v_1, v_2, \ldots, v_n\}$ is a basis for $V$, then the $j$-th column of $A$ is the coordinate vector of $T(v_j)$ with respect to the basis of $W$
• To find the image of a vector $v$ under $T$, multiply the matrix $A$ by the coordinate vector of $v$ with respect to the basis of $V$
  • If $v = c_1v_1 + c_2v_2 + \ldots + c_nv_n$, then $T(v) = A[v]_B = c_1T(v_1) + c_2T(v_2) + \ldots + c_nT(v_n)$, where $[v]_B$ is the coordinate vector of $v$ with respect to the basis $B$ of $V$

Dependence on choice of bases

• The matrix representation of a linear transformation depends on the choice of bases for the domain and codomain
  • Different bases for $V$ and $W$ will result in different matrix representations for the same linear transformation $T$
• Changing bases for the domain and codomain leads to similarity transformations of the matrix representation
  • If $A$ is the matrix of $T$ with respect to bases $B$ and $C$, and $P$ and $Q$ are change of basis matrices from $B$ to $B'$ and $C$ to $C'$, respectively, then the matrix of $T$ with respect to bases $B'$ and $C'$ is $Q^{-1}AP$

Matrices of linear transformations with respect to bases

Computing the matrix of a linear transformation

• To find the matrix of a linear transformation $T$ with respect to given bases $B$ and $C$, apply $T$ to each basis vector in $B$ and express the result as a linear combination of the basis vectors in $C$
• The coefficients of these linear combinations form the columns of the matrix representation of $T$ with respect to the bases $B$ and $C$
  • If $B = \{v_1, v_2, \ldots, v_n\}$ is a basis for $V$ and $C = \{w_1, w_2, \ldots, w_m\}$ is a basis for $W$, and $T(v_j) = a_{1j}w_1 + a_{2j}w_2 + \ldots + a_{mj}w_m$, then the matrix of $T$ with respect to $B$ and $C$ is $A = (a_{ij})$

Composition of linear transformations and matrix multiplication

• The matrix of a composition of linear transformations is the product of their individual matrices, with the order of multiplication determined by the order of composition
  • If $T: U \to V$ and $S: V \to W$ are linear transformations with matrices $A$ and $B$ with respect to bases of $U$, $V$, and $W$, respectively, then the matrix of $S \circ T$ with respect to the same bases is $BA$
• Matrix multiplication is associative and distributive over matrix addition, mirroring the properties of composition of linear transformations

Kernel and range of linear transformations

Kernel (null space) of a linear transformation

• The kernel (or null space) of a linear transformation $T$ is the set of all vectors $v$ in the domain such that $T(v) = 0$
  • $\ker(T) = \{v \in V \mid T(v) = 0\}$
• The kernel is a subspace of the domain and its dimension is called the nullity of the transformation
  • The kernel is closed under vector addition and scalar multiplication, making it a subspace
  • The nullity of $T$ is denoted by $\text{nullity}(T) = \dim(\ker(T))$
• Finding the kernel of a linear transformation represented by a matrix $A$ is equivalent to solving the homogeneous system of linear equations $Ax = 0$

Range (image) of a linear transformation

• The range (or image) of a linear transformation $T$ is the set of all vectors $w$ in the codomain such that $w = T(v)$ for some vector $v$ in the domain
  • $\text{range}(T) = \{w \in W \mid w = T(v) \text{ for some } v \in V\}$
• The range is a subspace of the codomain and its dimension is called the rank of the transformation
  • The range is closed under vector addition and scalar multiplication, making it a subspace
  • The rank of $T$ is denoted by $\text{rank}(T) = \dim(\text{range}(T))$
• Finding the range of a linear transformation represented by a matrix $A$ is equivalent to finding the column space of $A$

Rank-nullity theorem

• The rank-nullity theorem states that for a linear transformation $T$ from an $n$-dimensional vector space to an $m$-dimensional vector space, the sum of the rank and nullity of $T$ is equal to $n$
  • $\text{rank}(T) + \text{nullity}(T) = n$, where $n = \dim(V)$
• This theorem relates the dimensions of the kernel and range of a linear transformation to the dimension of its domain
• The rank-nullity theorem is a powerful tool for understanding the properties of linear transformations and their matrix representations