Final answer:
The statement is true; a 3×2 matrix cannot map the 2-dimensional vector space R^2 onto the 3-dimensional vector space R^3 because it can at most span a 2-dimensional subspace in R^3.
Step-by-step explanation:
If A is a 3×2 matrix, then the transformation x→Ax refers to a linear transformation that maps a 2-dimensional vector space (R^2) into a 3-dimensional vector space (R^3). The statement 'If A is a 3×2 matrix, then the transformation x→Ax cannot map R^2 onto R^3' is True. This is because a 3×2 matrix has only two columns, which correspond to the base vectors of R^2. It can only produce a span of at most two dimensions in R^3, since any vector in the image of the transformation is a linear combination of these two columns. As a result, the span of the vectors generated by this transformation cannot fill the entire 3-dimensional space.
It's also true that every 2-D vector can be expressed as the product of its x and y-components, and a vector can indeed form the shape of a right angle triangle with its x and y components, supporting the idea of vectors being expressed in terms of components.