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  1. #1
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    Basis and dimension question

    Another basis questions I'm having trouble with:
    Find basis for the following subsbace of R^5:
    W1 = {(a1,a2,a3,a4,a5) which is an element of R^5: a1-a3-a4=0}
    and
    W2 = {(a1,a2,a3,a4,a5) which is an element of R^5: a2=a3=a4 and a1+a5=0}

    What are the dimensions of W1 and W2?

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by lllll View Post
    Another basis questions I'm having trouble with:
    Find basis for the following subsbace of R^5:
    W1 = {(a1,a2,a3,a4,a5) which is an element of R^5: a1-a3-a4=0}
    and
    W2 = {(a1,a2,a3,a4,a5) which is an element of R^5: a2=a3=a4 and a1+a5=0}

    What are the dimensions of W1 and W2?

    Thanks in advance.
    A typical element of W1 is (a3 + a4, a2, a3, a4, a5) (why?). One possible basis is therefore {(1, 0, 1, 0, 0), (1, 0, 0, 1, 0), (0, 1, 0, 0, 0), (0, 0, 0, 0, 1)}. Check that it spans and is linearly independent. The dimension of W1 is the number of vectors in its basis ......

    A typical element of W2 is (a1, a2, a2, a2, -a1) (why?). One possible basis is therefore {(1, 0, 0, 0, -1), (0, 1, ,1 ,1, 0)}. Check that it spans and is linearly independent. The dimension of W2 is the number of vectors in its basis ...
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    Quote Originally Posted by mr fantastic View Post
    A typical element of W1 is (a3 + a4, a2, a3, a4, a5) (why?). One possible basis is therefore {(1, 0, 1, 0, 0), (1, 0, 0, 1, 0), (0, 1, 0, 0, 0), (0, 0, 0, 0, 1)}. Check that it spans and is linearly independent. The dimension of W1 is the number of vectors in its basis ......

    A typical element of W2 is (a1, a2, a2, a2, -a1) (why?). One possible basis is therefore {(1, 0, 0, 0, -1), (0, 1, ,1 ,1, 0)}. Check that it spans and is linearly independent. The dimension of W2 is the number of vectors in its basis ...
    How did you actually get to {(1, 0, 1, 0, 0), (1, 0, 0, 1, 0), (0, 1, 0, 0, 0), (0, 0, 0, 0, 1)}?
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    Quote Originally Posted by lllll View Post
    How did you actually get to {(1, 0, 1, 0, 0), (1, 0, 0, 1, 0), (0, 1, 0, 0, 0), (0, 0, 0, 0, 1)}?
    Look at (a3 + a4, a2, a3, a4, a5) and think of a simple way of constructing it .......
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