I am taking a class in Algebra but I am having a problem grasping exactly what it is I am being asked to do -- I think I am having a problem with the vocabulary being used. I have a couple of questions below and I would appreciate it, if you could take time to explain(in layman's terms or maybe programmer terms, exactly what is being asked. thanks in advance.
1) Let $[e_1; e_2; e_3]$ be the standard basis vectors in $\mathbb{R}^{3}$ and consider the ordered basis: $[e_2; e_1; e_3 + e_1]$ Verify that this is actually a basis and find the coordinates of the vector $(1; 1; 1)^T$ with respect to that basis.
2) Let $T$ be the linear map from $\mathbb{R}^{2}$ to $\mathbb{R}$ defined by: $T ((x, y)^T ) = x-y$ Find its matrix (with respect to the standard bases) and a basis for its kernel
3) Find a spanning vector set for the image of the linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{3}$ defined by: $T ((x, y)^T) = (2x-y, x + y, y)$
4) Consider the subspace U of $\mathbb{R}^3$ defined by: $ U = \{(x; y; z)^T: 2x- y + z = 0; x + y = 0 \} $. Express U as the kernel of an appropriately defined linear map and the matrix of that map with respect to the standard bases of the corresponding $\mathbb{R}^n$'s