Final answer:
The true statements are: If T is a linear transformation, then T(−u)=T(u);If T is a linear transformation, then T(0)=0;The range of T is Rᵐ;A vector b in Rᵐ is in the range of T if and only if Ax=b has a solution.
Step-by-step explanation:
In this question, we are given a transformation T:Rⁿ→Rᵐ defined by T(x)=Ax. Let's go through each option to determine which are true:
1) If T is a linear transformation, then T(-u) = T(u).
This statement is true because for a linear transformation, applying it to the negative of a vector gives the same result as applying it to the original vector.
2) If T is a linear transformation, then T(0) = 0.
This statement is also true because the zero vector in the domain maps to the zero vector in the range for a linear transformation.
3) The range of T is Rᵐ.
This statement is true because the range of a linear transformation is a subspace of the codomain, which in this case is Rᵐ.
4) A vector b in Rᵐ is in the range of T if and only if Ax=b has a solution.
This statement is true because a vector b will be in the range of T if and only if there exists a vector x such that Ax=b.
5) The domain of T is Rⁿ.
This statement is also true because T is defined from Rⁿ to Rᵐ.
Therefore, the correct answers are options 1, 2, 3, and 4.