Question 1. Every matrix transformation is a linear transformation.
Answer: True.
Question 2. If T is a map from a real n-dimensional space to a real m-dimensional space and if c is a vector in the m-dimensional space, then a uniqueness question is, "Is c in the range of T?"
Answer: False, since 0v1 + 0v2 + 1v3 + 0 v4 = 0.
Question 3.A linear transformation preserves the operations of vector addition and scalar multiplication.
Answer: True.
Question 4. A linear transformation from R^n to R^m is completely determined by its effect on the columns of the identity matrix I_n.
Answer: True.
Question 5. A mapping T:R^n -> R^m is one-to-one if each vector in R^n maps onto a unique vector in R^m.
Answer: False. This is true of every mapping T. The one-to-one part is the other direction. Does every vector in the image come from exactly one vector. Or in other words, is every vector in R^m the image of either 0 or 1 vector?