(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let W be a subspace of V = [tex]C^5[/tex] consisting of all vectors [tex]

x = (x_1,x_2,x_3,x_4,x_5) \epsilon C^5[/tex] that satisfy:

[tex]

-2ix_1 + x_2 - x_3 + (1 - i)x_4 = 0

[/tex]

[tex]

x_1 + ix_2 -2x_5 = 0

[/tex]

Find a set that spans W.

2. Relevant equations

3. The attempt at a solution

From the second equation we know that [tex] x_1 = 2x_5 - ix_2[/tex]. Substituting that into the first, we have [tex]-2i(2x_5 -ix_2) + x_2 - x_3 + (1-i)x_4 = 0[/tex]. Expanding through the brackets and simplifying, we have [tex]-4ix_5 - x_2 - x_3 + (1-i)x_4 = 0[/tex]. This implies that [tex]x_2 = -x_3 +(1-i)x_4 -4ix_5[/tex]. So that is the only condition we have. So, we now have the following:

[tex](1, 0, 0, 0, 0)x_1 + (0, -1, 1, 0, 0)x_3 + (0, 1-i, 0, 1, 0)x_4 + (0, -4i, 0, 0, 1)x_5[/tex]

So the set [tex]{ (1, 0, 0, 0, 0), (0, -1, 1, 0, 0), (0, 1-i, 0, 1, 0), (0, -4i, 0, 0, 1)[/tex] should span the space. However, if you multiply the first vector by 1, and the rest by 0, we have [tex](1,0,0,0,0)[/tex], which obviously doesn't satisfy the two equations above.

What am I doing wrong?

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# Homework Help: Finding a spanning set

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