How do we show that a given set is a subspace of C^3?

I have been dwelling on this problem for hours now. Can someone please help me?





Decide whether the given set S is a subspace of C^3.





S =


( a - b )


( 3ib )


( (2 + i)a - 2b )





where a, b are any complex numbers.

How do we show that a given set is a subspace of C^3?
To show that a subset is a subspace, you must show simply that for any x,y in the set and a,b scalars, you have that ax+by is in the subset.





Unfortunately, I don't understand your notation for your sets. Is that supposed to represent vectors of the form %26lt;a-b, 3ib, (2+i)a -2b%26gt;?





If so, let's suppose x and y are of that format, and p and q are scalars. Let x be determined by complex numbers a1 and b1, and y be determined by complex numbers a2 and b2. Then we have that


px + qy = %26lt;p(a1-b1) + q(a2-b2), 3ipb1 + 3iqb2, (2+i)pa1 - 2pb1 + (2+i)qa2 - 2qb2%26gt;


= %26lt;(pa1 + qa1) - (pb1 + qb2), 3i(pb1+qb2), (2+i)(pa1+qa2) - 2(pb1+qb2)%26gt;,


which is of the specified form with


a = pa1+qa2,


b = pb1+qb2.





Therefore, S is a subspace.


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