M115A-DirectSumOfSubspaces

M115A-DirectSumOfSubspaces - Direct Sum of Subspaces Paul...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Direct Sum of Subspaces Paul Skoufranis September 29, 2011 The purpose of this document is to demonstrate that vector spaces can have ’nice’ decompositions. In particular, we will present some proofs relating to spaces of vector spaces. We begin with a definition for the sum of two sets of vectors. Definition) Let V be a vector space. Suppose that S 1 and S 2 are non-empty subsets of V . The sum of S 1 and S 2 , denoted S 1 + S 2 , is { ~x + ~ y | ~x S 1 , ~ y S 2 } (that is, S 1 + S 2 is the set of vectors of V that can be obtained by adding a vector in S 1 to a vector in S 2 ). Example) Let S 1 be the x -axis in R 3 and let S 2 be the y -axis in R 3 . Then S 1 + S 2 is the xy -plane in R 3 . To see this, we notice that if we add a vector in the direction of the x -axis to a vector in the direction of the y -axis, we obtain a vector in the xy -plane (that is ( x, 0 , 0) + (0 , y, 0) = ( x, y, 0) is in the xy -plane). Moreover, every vector in the xy -plane can be written as the sum of a vector in the direction of the x -axis and a vector in the direction of the y -axis (that is, ( x, y, 0) = ( x, 0 , 0) + (0 , y, 0)). Example) Let S 1 = { ( x, 0) | x R } ⊆ R 2 and S 2 = { ( y, y ) | y R } ⊆ R 2 } . Then S 1 + S 2 R 2 (as span { (1 , 0) , (1 , 1) } = R 2 ). The following is our first result about the sum of two subspaces of a vector space. Proposition) Let V be a vector space and suppose W 1 and W 2 are subspaces of V . Then W 1 + W 2 is a subspace of V that contains W 1 and W 2 . Proof : Clearly W 1 + W 2 V . To show that W 1 + W 2 is a subspace of V , we need to demonstrate the three properties of being a subspace. 1. We need to demonstrate that ~ 0 W 1 + W 2 . We recall that since W 1 and W 2 are subspaces of V , ~ 0 W 1 and ~ 0 W 2 . Therefore, since ~ 0 = ~ 0 + ~ 0, ~ 0 W 1 , and ~ 0 W 2 , ~ 0 W 1 + W 2 by the definition of W 1 + W 2 . 2. We need to demonstrate that W 1 + W 2 is closed under addition. Suppose ~x, ~ y W 1 + W 2 are arbitrary vectors. Therefore, by the definition of W 1 + W 2 , there exists vectors ~x 1 , ~ y 1 W 1 and ~x 2 , ~ y 2 W 2 such that ~x = ~x 1 + ~x 2 and ~ y = ~ y 1 + ~ y 2 . Therefore ~x + ~ y = ( ~x 1 + ~x 2 ) + ( ~ y 1 + ~ y 2 ) = ( ~x 1 + ~ y 1 ) + ( ~x 2 + ~ y 2 ) Since ~x 1 , ~ y 1 W 1 and W 1 is a subspace of V , ~x 1 + ~ y 1 W 1 . Since ~x 2 , ~ y 2 W 2 and W 2 is a subspace of V , ~x 2 + ~ y 2 W 2 . Therefore, we have shown that ~x + ~ y is the sum of a vector in W 1 and a vector in W 2 . Hence ~x + ~ y W 1 + W 2 by the definition of W 1 + W 2 . Therefore, since ~x, ~ y W 1 + W 2 were arbitrary, W 1 + W 2 is closed under addition.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern