Problem 5: Suppose V is finite-dimensional and P € L(V) is such that P² = P and that every vector in ker(P) is orthogonal to every vector in range(P). Show that there exists a subspace U of V such that P = Pru.
Problem 5: Suppose V is finite-dimensional and P € L(V) is such that P² = P and that every vector in ker(P) is orthogonal to every vector in range(P). Show that there exists a subspace U of V such that P = Pru.
Problem 5: Suppose V is finite-dimensional and P € L(V) is such that P² = P and that every vector in ker(P) is orthogonal to every vector in range(P). Show that there exists a subspace U of V such that P = Pru.
Transcribed Image Text:Problem 5: Suppose V is finite-dimensional
and P € L(V) is such that P²
P
and that every vector in ker(P) is orthogonal to every vector in range(P). Show that
there exists a subspace U of V such that P = Pru.
=
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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