Let T be a linear transformation. If we have this then what statements would hold true?
- T contains $\vec{0}$ (i.e $\vec{0} \in T$)
- T is a 3 x 4 matrix
- The columns of T are linearly independent
- T($\vec{0}$) = $\vec{0}$
- T is one-to-one
- T is non-empty
- The domain of T is $\mathbb{R}^4$
These sort of questions are given to us after each topic in algebra to understand if we know the concepts inside out.
My attempt would be that 3, 4 and 5 are true. Im unsure about others. 3, 4 and 5 I was able to find because of a theorem. I think 1 should be true, since $\vec{0}$ should be in the vector space. 7 can be true or can be false. The question doesn't specify so we assume it to be false. 2 and 6 i have no idea at all. Can T be empty? Because it would make no sense for it to be called a transformation if its empty.
Appreciate your time and effort
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