Final answer:
The statement is FALSE. A linear transformation is one-to-one if there is a pivot position in every column, not row. T: R³ → R² cannot be one-to-one as it maps from a higher to a lower-dimensional space.
Step-by-step explanation:
If there is a pivot position in every row of the standard matrix A for the linear transformation T: R³ → R², is T one-to-one is the question being asked. This statement is FALSE. A linear transformation T is one-to-one if each input vector in the domain corresponds to a unique output vector in the codomain. In terms of the matrix representation of the transformation, T is one-to-one if and only if the matrix has a pivot position in every column when the matrix is in echelon form, which would imply that the transformation is injective. Since our matrix is mapping from R³ to R², it means we have a 2x3 matrix. For a transformation from a higher-dimensional space to a lower-dimensional space (3D to 2D in this case), it is inherently not possible for every vector in R³ to be uniquely mapped to R². This is because there are more vectors in R³ than there are in R², leading to unavoidable overlap when projecting down to a smaller space — that is, at least one vector in R³ must share an output with another, meaning T cannot be one-to-one.