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how do I check if the following set of vectors $[3+i,0,i],[0,1+3i,2],[6+2i,3+9i,6+2i]$, or in general any set of complex vectors span the space $\Bbb C^3$, not using any matrix methods?

Thanks for helping out!

Yonatan Izutskiver
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    Two vectors cannot span a three dimensional vector space. It takes at least three and they need to be linearly independent. For four or more vectors we need to be able to chose three of them that are linearly independent. – marwalix Nov 15 '15 at 10:46
  • @marwalix be careful about the dimension of the vector space in question. – frosh Nov 15 '15 at 10:49
  • @Mert Aktas The dimension depends on the base field. If we consider complex vector space (base field $\mathbb{C}$) the dimension of $\mathbb{C}^3$ is three. If it is a real vector space (base field $\mathbb{R}$) dimension is six.... – marwalix Nov 15 '15 at 11:04

2 Answers2

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A set of $n$ vectors span a $n$-dimensional vector space if this set is a basis, and this is done if the $n$ vectors are linearly independent.

In your case you have three vectors $$ v_1=[3+i,0,i]\quad v_2=[0,1+3i,2]\quad v_3=[6+2i,3+9i,6+2i] $$ that are linearly dependent if there exist three scalars not all null $\alpha,\beta,\gamma$ such that $ \alpha v_1+\beta v_2+\gamma v_3=0$

In general we can find these numbers solving a system of equation (in this case three equations in three unknowns) and this can be done , if we want, without matrix calculations.

Note that this three numbers are scalars of the vectors space so, in general, it is important to specify the field over which the vector space is defined. In you case, $\mathbb{C}^3$ can be a vector space over $\mathbb{Q}$ or $\mathbb{R}$ or $\mathbb{C}$, and the answer can, in general, be different ( note that such vector spaces have different dimension).

For the given vectors the answer is aniway simple since we easily see that: $$ 2 v_1+3v_2=v_3 $$ so the three vectors are linearly dependent and they are not a basis for any of the possible vector spaces.

Emilio Novati
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First you should determine the dimension of the vector space $\Bbb C^3$.

Then, remember this: if the desired vector space has a dimension $n$, you need at least $n$ linearly independent vectors to span this vector space. Thus, you do not need any matrix knowledge here, because $\Bbb C^3$ 's dimension is greater than $3$.

frosh
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