Linear Transformations and Matrices

GRE Math Subject · Axiom Academy

LESSON Linear Transformations and Matrices Connecting abstract transformations to concrete matrix representations A function between vector spaces is a linear transformation if: Matrix of a Linear Transformation Given bases of and of , the matrix representation is: Formula: If is the coordinate vector of in basis , then: Note: Different bases give different matrix representations, but they represent the same linear transformation. Bijective (isomorphism) if both conditions hold For compositions , the matrix is the product: Example: For a matrix, if rank is 3, then nullity is . is bijective (both injective and surjective) The matrix is invertible (determinant nonzero) GRE Tip: Questions frequently ask about kernel, image, or invertibility. Always compute the rank to determine if a transformation is injective.

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