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Direct Sum Decomposition

In each of the problems with square check boxes, you need to check all the correct answers in order for the answer to be marked correct.

SKIP_ p1 QM_[0;true;true;false;true] In each case, decide if it is true that images/ma322c2d1.png is the direct sum of the subspaces U and V. AH_[2]
  1. U = {(a, 0) | a real} and V = {(a, -a) | a real} AS_[true;false]
  2. U = {(a, -a) | a real} and V = {(a, a) | a real} AS_[true;false]
  3. U = {(a, -a) | a real} and V = {(a-1;1-a) | a real} AS_[true;false]
  4. U = {(a, b) | a, b real} and V = {(0, 0)} AS_[true;false]
SKIP_ p2 QM_[0;is;is;is;is not] Consider the subspaces U = {(a, 0) | a real}, V = {(0,b) | b real} and W = {(c, c) | c real} of images/ma322c2d2.png . AH_[2]
  1. The sum of any two of the subspaces AS_[is;is not] equal to the vector space images/ma322c2d3.png .
  2. The sum of all three of the subspaces AS_[is;is not] equal to the vector space images/ma322c2d4.png .
  3. The intersection of any two of the subspaces AS_[is;is not] the trivial subspace consisting of just {(0, 0)}.
  4. The vector space images/ma322c2d5.png AS_[is;is not] the direct sum of U, V, and W.
SKIP_ p3 QM_[0;true;true;true;true;true;true] In this problem, we will be working in the vector space of all real valued functions defined for all real numbers. Recall that such a function f(x) is said to odd if f(-x) = -f(x) for all real x, and that it is said to be even if f(-x) = f(x) for all real x. For example, the sine function is odd and the cosine function is even. Indicate which of the following are true; make sure you can either prove each assertion or give a counter-example. AH_[2]
  1. If f(x) is any real valued function defined for all real numbers, then the function f(x) - f(-x) is an odd function. AS_[true;false]
  2. If f(x) is any real valued function defined for all real numbers, then the function f(x) + f(-x) is an even function. AS_[true;false]
  3. The set U of odd real valued functions forms a vector subspace. AS_[true;false]
  4. The set V of even real valued functions forms a vector subspace. AS_[true;false]
  5. The vector space of all real valued functions is the sum of U and V AS_[true;false]
  6. The vector space of all real valued functions is the direct sum of U and V AS_[true;false]
SKIP_ p4 QM_[0;true;true;true;true;true] Consider the vector space M of m by m square matrices. Recall that a square matrix images/ma322c2d6.png is said to be symmetric if images/ma322c2d7.png for all indices i and j. The same matrix is said to be antisymmetric if images/ma322c2d8.png for all indices i and j. Indicate which of the following are true; be sure that you can either prove the assertion or give a counter-example. AH_[2]
  1. If images/ma322c2d9.png is anti-symmetric, then images/ma322c2d10.png for all indices i. AS_[true;false]
  2. The set U of symmetric m by m matrices is a vector subspace of M. AS_[true;false]
  3. The set V of antisymmetric m by m matrices is a vector subspace of M. AS_[true;false]
  4. M is the sum of U and V. AS_[true;false]
  5. M is the direct sum of U and V. AS_[true;false]
SKIP_ p5 QM_[0;true] Let U be any subspace of a finite dimensional vector space V. Extend a basis images/ma322c2d11.png of U to a basis images/ma322c2d12.png of V. Let W be the span of images/ma322c2d13.png . Then V is the direct sum of U and W AS_[true;false] Make sure you can either prove the assertion of give a counterexample. SKIP_ p6 QM_[0;true;true] Let U and W be arbitrary subspaces of a vector space V. Let images/ma322c2d14.png be a basis for the subspace images/ma322c2d15.png . Extend this to a basis images/ma322c2d16.png of U. Extend the basis again to a basis images/ma322c2d17.png of W. AH_[2]
  1. The set images/ma322c2d18.png is a basis for U + W. AS_[true;false]
  2. dim(U + W) = dim(U) + dim(W) - dim(Uimages/ma322c2d19.png V). AS_[true;false]
Make sure you can prove your assertions. SKIP_ p7 QM_[0;yes] Two subspaces U and W of images/ma322c2d20.png are said to be perpendicular to each other if the dot product of any vector in U with any vector in V is always zero. Is it true that, if U and W are perpendicular, then U + W is the direct sum of U and W? Why? AS_[yes;no] SKIP_ p8 QM_[0;no;no] Let U, V, and W be subspaces of vector space such that U + W = V + W. AH_[2]
  1. Does it follow that U = V? AS_[yes;no]
  2. Does it follow that U = V if the sum is direct? AS_[yes;no]
Make sure you can either prove the assertion or give a counter-example. SKIP_ p9 QM_[0;no] Let U, V, and W be subspaces of a vector space. Suppose that for every two of these subspaces, the sum of the two spaces is direct. Does it follow that the sum of all three is direct? AS_[yes;no] Make sure you can either prove the assertion or give a counter-example. SKIP_