1.2 Linear transformations and matrices

Linear transformations are the backbone of vector space operations. They preserve key properties like addition and scalar multiplication, allowing us to map vectors between spaces. These transformations are crucial for understanding how vectors behave under different operations.

Matrices provide a concrete way to represent and work with linear transformations. By encoding the transformation's effect on basis vectors, we can use matrix operations to combine, scale, and analyze transformations. This approach connects abstract concepts to practical problem-solving in linear algebra.

Linear Transformations

Properties of linear transformations

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• Linear transformations map vectors between spaces preserving addition and scalar multiplication $T: V \rightarrow W$ (V and W are vector spaces)
• Additivity property ensures $T(u + v) = T(u) + T(v)$ for all vectors u and v in V
• Homogeneity property maintains $T(cv) = cT(v)$ for all vectors v in V and scalars c
• Common examples include rotation, reflection, scaling, and projection (matrix multiplication)
• Non-linear transformations include translation and quadratic functions (polynomial transformations)

Matrix representation of transformations

• Standard basis vectors form foundation for matrix representation
• Transformation matrix encodes linear map's action on basis vectors
• Process to find matrix representation:
  1. Apply transformation to each basis vector
  2. Write resulting vectors as columns of the matrix
• Matrix operations mimic linear transformation behavior:
  • Multiplication combines transformations
  • Addition superimposes transformations
  • Scalar multiplication scales transformation effect
• Matrices solve linear equation systems via Gaussian elimination or LU decomposition
• Transformation analysis utilizes determinant, eigenvalues, eigenvectors, and trace

Kernel and range of transformations

• Kernel (nullspace) contains vectors mapping to zero vector $\text{ker}(T) = \{v \in V : T(v) = 0\}$
• Range (image) encompasses all possible output vectors $\text{range}(T) = \{w \in W : w = T(v) \text{ for some } v \in V\}$
• Matrix nullspace solves $Ax = 0$, corresponding to transformation kernel
• Matrix column space spans column vectors, equivalent to transformation range
• ###rank-nullity_theorem_0### links domain dimension to kernel and range dimensions $\dim(\text{ker}(T)) + \dim(\text{range}(T)) = \dim(V)$

Composition of linear transformations

• Composition applies transformations sequentially $(T_2 \circ T_1)(v) = T_2(T_1(v))$
• Matrix representation of composition $BA$ represents $T_2 \circ T_1$ (A for $T_1$, B for $T_2$)
• Composition properties include associativity but generally not commutativity
• Identity transformation acts as neutral element in composition $I \circ T = T \circ I = T$
• Inverse transformations undo original transformation $T^{-1} \circ T = T \circ T^{-1} = I$
• Composition analysis reveals $\text{ker}(T_2 \circ T_1) \supseteq \text{ker}(T_1)$ and $\text{range}(T_2 \circ T_1) \subseteq \text{range}(T_2)$