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Linear Transformations - Deep Dive

Mathematical Foundations​

Rigorous treatment of linear transformations with proofs and advanced theory.

Formal Definitions​

Linear Transformation​

Definition: A function T: V → W between vector spaces is linear if...

Matrix Representation Theorem​

Theorem: Every linear transformation T: ℝⁿ → ℝᵐ can be represented as...

Proof: (To be completed)

Kernel and Image​

Fundamental Theorems​

Rank-Nullity Theorem: For linear T: V → W, dim(V) = dim(ker(T)) + dim(im(T))

Proof: (To be completed)

Properties​

(To be completed)

Composition and Invertibility​

Theorems​

Theorem: The composition of linear transformations is linear.

Proof: (To be completed)

Change of Basis​

Theory​

(Mathematical treatment to be completed)

Exercises​

(Advanced problems to be completed)