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Math Help - help understanding basis and dimension

  1. #1
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    help understanding basis and dimension

    I am working on the following two problems:

    Consider the subset U = {(x1,4x1-7x2,x2) : x1, x2 are real numbers} of R^3

    Assume that U is a subspace of R^3.

    Find a basis for the subspace U = {(x1,4x1-7x2,x2) : x1, x2 are real numbers} of R^3 and explain why it is a basis for U.

    I have deduced that U has the correct number of elements which is 3.

    I need to show that U spans R^3 and that U is linearly independant.

    However I am for some reason having a hard time in doing so... I'm pretty sure this is all I need to do? But just can't get this problem started.


    Part b) asks to find the dimension of said subspace. The dimension (according to my notes) of R^n with standard operations is n. So in this case it is 3 right ?
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  2. #2
    Senior Member Tinyboss's Avatar
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    U is the image of a linear map from R2 to R3 given by f(x_1,x_2)=(x_1,4x_1-7x_2,x_2). It's pretty easy to see that f is zero only at (0,0), (i.e. f has trivial kernel), so that the image (U) has the same dimension as the domain (R2). Whenever that happens, you can take a basis of the domain, and the image of the basis set forms a basis of the image. So {f(1,0), f(0,1)} is a basis for U.
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  3. #3
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    Quote Originally Posted by battleman13 View Post
    I am working on the following two problems:

    Consider the subset U = {(x1,4x1-7x2,x2) : x1, x2 are real numbers} of R^3

    Assume that U is a subspace of R^3.

    Find a basis for the subspace U = {(x1,4x1-7x2,x2) : x1, x2 are real numbers} of R^3 and explain why it is a basis for U.
    Any vector in that subspace is of the form (x_1, 4x_1- 7x_2, x_2)= (x_1, 4x_1, 0)+ (0, -7x_2, x_2)= x_1(1, 4, 0)+ x_2(0, -7, 1).

    That tells you the basis and dimension immediately.

    [I have deduced that U has the correct number of elements which is 3.

    I need to show that U spans R^3 and that U is linearly independant.

    However I am for some reason having a hard time in doing so... I'm pretty sure this is all I need to do? But just can't get this problem started.


    Part b) asks to find the dimension of said subspace. The dimension (according to my notes) of R^n with standard operations is n. So in this case it is 3 right ?
    No, it isn't! The dimension of R^3 is 3 but this is a subspace of R^3. The fact that every vector in the subspace depended upon two numbers, x_1, and x_2 should hve been a clue!
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