Question

A linear transformation $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is *one-to-one* if for all $\vec{b} \in \mathbb{R}^{m}$ there is at most one (possibly no) $\vec{x} \in \mathbb{R}^{n}$ so that $T(\vec{x}) = \vec{b}$

Answer 1

A linear transformation is one-to-one if there is at most one solution for each output vector. To check if a linear transformation is one-to-one, we can solve the equation T(x, y, z) = (0, 0). If the only solution is (0, 0, 0), then the transformation is one-to-one.

Step-by-step explanation:

A linear transformation is one-to-one if there is at most one solution for each output vector. In other words, for any vector b in the range of the transformation, there is at most one vector x in the domain such that T(x) = b. This means that every output vector is uniquely determined by its corresponding input vector.

To determine if a linear transformation is one-to-one, we can check if the null space of the transformation contains only the zero vector. If the null space contains any vectors other than the zero vector, the transformation is not one-to-one.

For example, consider the linear transformation T : R^3 -> R^2 defined by T(x, y, z) = (2x + y, x - z). To check if it is one-to-one, we solve the equation T(x, y, z) = (0, 0). If the only solution is (0, 0, 0), then the transformation is one-to-one.