Basis and dimension on complex vector spaces

Sep 2012
Find the dimension of and a basis for the following subspaces of C^3:
(i) The set of all complex multiples of (1,i,1-i)

(ii) The plane z1+z2+(1-i)z3=0

(iii) The range of the matrix A with rows (1,i,2-i) and (2+i, 1+3i, -1-i)

(iv) The kernel of the same matrix.

(v) The set of vectors that are orthogonal to (1-i, 2i, 1+i).

Please help. I am a bit confused.

Thank you
Sep 2012
Washington DC USA
*Presumably*, given your title and the implicit assumption when things are stated this way,

by the vector space \(\displaystyle \mathbb{C}^3\), you mean the vector space over the base field \(\displaystyle \mathbb{C}.\)

For (i) maybe this is a case where abstraction clarifies rather than confuses? Maybe it removes the the jumble formed by the specific values?
Suppose I tell you I have a vector space V over a field F. Suppose I give you a fixed vector in V, and ask you to find a basis and the dimension of the subspace formed by all F-multiples of that vector:

\(\displaystyle \text{Let } V \text{ be a vector space over field } \mathbb{F}.\)

\(\displaystyle \text{Let } \vec{v_0} \in V - \{ \vec{0} \}. \text{ Let } W = \{ a\vec{v_0} \in V | a \in \mathbb{F} \}.\)

\(\displaystyle \text{What is the dimension of the subspace } W? \text{ What is a basis for } W?\)

You do everything *exactly* as you would for a real vector space, except now the coefficients are allowed to be complex numbers. That goes not just for problem (i), but for all 5 problems. Ask yourself, "How would I do the same problem, except for a real vctor space using real coefficients?" If you can answer that, then you should be able to solve these. Every procedure is *exactly* the same, except that here it uses complex coefficients.
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