Linear Algebra: Matrix Transformations and Linear Maps

For the transformation, identify all vectors that fall on the line $y = -2x$ and are transformed to the zero vector, and describe the image set which consists of vectors that land on the line $y = x$.

Consider the linear transformation that maps $\mathbb{R}^3$ to $\mathbb{R}^2$ given by $L(v) = (v_1, v_2 - v_3)$. Find the image of the subspace of $\mathbb{R}^3$ given by the vectors of length 3 where the second element is two times the first element, and the third element is 0.

Consider the linear transformation that maps $\mathbb{R}^3$ to $\mathbb{R}^2$ given by $L(v) = (v_1, v_2 - v_3)$. Find the kernel of this transformation.

Find the kernel and range of the differentiation operator on $P_2$. Specifically, for the linear transformation $D$ from the vector space of polynomials of degree two or less ($P_2$) to the vector space of polynomials of degree one or less ($P_1$), determine the kernel and range.

Find an $X$ in $\mathbb{R}^2$ so that the image of $X$ is $\begin{pmatrix} 3 \\ 2 \\ -5 \end{pmatrix}$.

Is the vector $\begin{pmatrix} 3 \\ 2 \\ 5 \end{pmatrix}$ in the range of $T$?

Describe the effects of applying one transformation and then another, specifically using the example of first rotating the plane 90 degrees counterclockwise, then applying a shear. Determine the overall effect as a single linear transformation and express it with a matrix.