I have the following statements that I am supposed to determine whether are true or false

$C = \{(1, 2, 3)^{T}, (2, 3, 4)^{T}\}$ is a basis of a 2-dimensional subspace of $\mathbb{R^{3}}$

$D = \{(-7, 0, 0, 0)^{T}\}$ is a basis of a line through the origin of $\mathbb{R^{4}}$

For $C$ I am not sure what to do. I know how I can find out if some set of vectors are a basis for a space (They span the space and they are linearly independent), but I dont understand this question. Any suggestions?

For $D$ I think it's true for any $(a,b,c,0)$ ?


For $C$ you have to check, that these vectors are linearly independent. $D$ is trivially a basis of a one dimensional subspace because its generated by just one vector not equal to zero and the origin lies in every vectorspace.

EDIT: Why it is enough to check linearly independency for $D$: In the question the considered subspace is $V:=\operatorname{span}\{(1,2,3)^T,(2,3,4)^T\}$. By definition $\operatorname{span}$ is a vectorspace. Its dimension equals the amount of linearly independent generators, which should have been proven in your linear algebra course.


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