Thread: 2-dimensional subspace U of C^3.

1. 2-dimensional subspace U of C^3.

Can you please Give me an example of a 2-dimensional subspace U of C^3.

thanks

2. $\displaystyle \mathbb{C}$ is two dimensional since $\displaystyle \mathbb{C} = \{a+ ib \colon a,b\in\mathbb{R}\}$, which means that $\displaystyle \mathbb{C}$ is a vector space with two $\displaystyle \mathbb{R}$ entries.

It's clear that $\displaystyle \mathbb{C} \subset \mathbb{C} ^ {3}$

Now using the definition of subspace:
Since $\displaystyle 0 \in \mathbb{C}$
and $\displaystyle \mathbb{C}$ is closed under addition and scalar multiplication $\displaystyle \mathbb{C} = U$ is a two dimensional subspace of $\displaystyle \mathbb{C}^{3}$

3. Thanks for your help. Now how do i
write a basis for this subspace.

4. Well a basis is any linearly independent that spans the vector space. In the case of $\displaystyle \mathbb{C}$, no real constants a,b, will make $\displaystyle a + ib = 0$ unless a=b=0. So any ordered pair of real numbers will constitute a basis for $\displaystyle \mathbb{C}$

5. Originally Posted by charikaar
Can you please Give me an example of a 2-dimensional subspace U of C^3.

thanks
Your question is imprecise. First, if $\displaystyle C$ is the field $\displaystyle \mathbb{C}$, then it is unclear over what field you are considering $\displaystyle \mathbb{C}^3$ to be a vector space. For instance it's a 6-dimensional vector space over $\displaystyle \mathbb{R}$, but a 3-dimensional one over $\displaystyle \mathbb{C}$. You always need to specify the underlying field. Haven, above, assumed that the vector space is over $\displaystyle \mathbb{R}$. Check your question!

6. Question is:
Give an example of a 2-dimensional subspace of
$\displaystyle \mathbb{C}^3$

thanks

7. Noting Bruno's comment on fields. If we assume $\displaystyle \mathbb{C}^{3}$ is an $\displaystyle \mathbb{R}$ vector space, then my above answer is correct. However if we view $\displaystyle \mathbb{C}^{3}$ as a $\displaystyle \mathbb{C}$ vector space. Then an example of a two-dimensional vector space would be $\displaystyle \mathbb{C}^{2}$

I assume based on the vagueness of the question that you are in a Linear Algebra class and you have little or no understanding of what a field is (in the context of the class). So the interpretation of $\displaystyle \mathbb{C}^{3}$ as an$\displaystyle \mathbb{R}$ vector space, is probably the desired one.