Hello, can anyone give me some help with these excersises?

1. Give an example of a linear transformation that is injective but not surjective
2. Give an example of a linear transformation which kernel is not only the indentity element (Ov)
I really don't know how to make it, Please help

I'd be happy to help you with these exercises! Let's break them down one by one.

1. To give an example of a linear transformation that is injective but not surjective, we need to find a transformation that sends different input vectors to different output vectors (injective) but doesn't cover the entire output space (not surjective).

One example could be the transformation T: R^2 -> R^2 defined by T(x, y) = (2x, 2y). This transformation doubles both coordinates of any vector. It is injective because different input vectors (x, y) would always produce different output vectors (2x, 2y). However, this transformation only maps to vectors in the xy-plane, so it is not surjective.

2. To find an example of a linear transformation where the kernel is not only the zero vector (Ov), we need to find a transformation that maps some input vectors to the zero vector but also maps other vectors to non-zero vectors.

One example could be the transformation T: R^3 -> R^2 defined by T(x, y, z) = (x - y, y - z). This transformation subtracts consecutive coordinates of any vector. The kernel of this transformation contains all vectors (x, y, z) such that x - y = 0 and y - z = 0. This simplifies to x = y = z, which means the kernel is the set of all vectors that satisfy this condition. As this is not only the zero vector, the kernel is not trivial.

Remember, these examples are just illustrations, and there can be many possible transformations that satisfy the given conditions. It's always a good idea to experiment and come up with other examples to deepen your understanding of linear transformations.