By Sergei Treil

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Examples. You dealt with linear transformation before, may be without even suspecting it, as the examples below show. Example. Differentiation: Let V = Pn (the set of polynomials of degree at most n), W = Pn−1 , and let T : Pn → Pn−1 be the differentiation operator, ∀p ∈ Pn . T (p) := p Since (f + g) = f + g and (αf ) = αf , this is a linear transformation. Example. Rotation: in this example V = W = R2 (the usual coordinate plane), and a transformation Tγ : R2 → R2 takes a vector in R2 and rotates it counterclockwise by γ radians.

Suppose A is invertible. Then x = A−1 b solves the equation Ax = b. To show that the solution is unique, suppose that for some other vector x1 ∈ V Ax1 = b −1 Multiplying this identity by A from the left we get A−1 Ax = A−1 b, and therefore x1 = A−1 b = x. Note that both identities, AA−1 = I and A−1 A = I were used here. Let us now suppose that the equation Ax = b has a unique solution x for any b ∈ W . Let us use symbol y instead of b. We know that given y ∈ W the equation Ax = y Doesn’t this remind you of a basis?

Let BA = I and AC = I. Then BAC = B(AC) = BI = B. On the other hand BAC = (BA)C = IC = C, and therefore B = C. Suppose for some transformation B1 we have B1 A = I. Repeating the above reasoning with B1 instead of B we get B1 = C. Therefore the left inverse B is unique. The uniqueness of C is proved similarly. Definition. A matrix is called invertible (resp. left invertible, right invertible) if the corresponding linear transformation is invertible (resp. left invertible, right invertible). 1 asserts that a matrix A is invertible if there exists a unique matrix A−1 such that A−1 A = I, AA−1 = I.