Let T : V → V be a linear operator on a finite-dimensional vector space V , such that any subspace W of V satisfies T(W) ⊆ W. (a) Show that for each non-zero vector v ∈ V , there exists a scalar av depending on v such that T(v) = avv. (b) Suppose that {u, v} is a linearly independent set of vectors in V . Show that the scalars au and av in part (a) are equal. (Hint: Consider T(u + v).)

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.