Let’s analyze and solve the given mathematical problems step by step.

1. Task 1: Linear Transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$

• Given data:

  • $T(1,0) = (3,-1,5)$
  • $T(0,1) = (3,1,-1)$

  We are tasked with finding $T(1,1)$.

Solution:

By the linearity of $T$, we can write: $T(1,1) = T(1,0) + T(0,1)$ Substitute: $T(1,1) = (3,-1,5) + (3,1,-1)$ Perform component-wise addition: $T(1,1) = (6,0,4)$

2. Task 2: Linear Transformation $T: \mathbb{R}^3 \to \mathbb{R}^2$

• Given transformation:
  $T(x,y,z) = (x + z, y + z)$
• Tasks:
  • Find the kernel, image, nullity, and rank of $T$.

Solution:

• Kernel:
  The kernel is the set of vectors $(x, y, z) \in \mathbb{R}^3$ such that $T(x,y,z) = (0,0)$.
  Solve: $x + z = 0 \quad \text{and} \quad y + z = 0$ From the first equation, $z = -x$.
  From the second equation, $z = -y$.
  Therefore, $x = y$, and the kernel is: $\text{Ker}(T) = \text{span}\{(1,1,-1)\}$

• Image:
  The image of $T$ is all possible outputs $(x+z, y+z)$. By varying $x, y, z$, we see the image is all of $\mathbb{R}^2$: $\text{Im}(T) = \mathbb{R}^2$

• Nullity:
  The nullity is the dimension of the kernel, which is 1 (since the kernel is spanned by one vector).

• Rank:
  The rank is the dimension of the image, which is 2 (since the image is $\mathbb{R}^2$).

• Verify Sylvester's Law:
  Using the rank-nullity theorem: $\text{Rank} + \text{Nullity} = \text{Dimension of domain}$ Here: $2 + 1 = 3$ Verified.

3. Task 3: Verify Linearity for $T: V_2 \to V_2$, $T(x,y) = (x+y, x+2)$:

• Check additivity:
  Let $u = (x_1, y_1)$, $v = (x_2, y_2)$.
  Check: $T(u + v) = T((x_1 + x_2, y_1 + y_2)) = ((x_1+x_2)+(y_1+y_2), (x_1+x_2)+2)$ On the other hand: $T(u) + T(v) = (x_1+y_1, x_1+2) + (x_2+y_2, x_2+2)$ Both simplify to: $((x_1+x_2)+(y_1+y_2), (x_1+x_2)+2)$ Hence, additivity is satisfied.

• Check homogeneity:
  Let $c$ be a scalar. Check: $T(cu) = T(c(x_1, y_1)) = T((cx_1, cy_1)) = (cx_1+cy_1, cx_1+2)$ On the other hand: $cT(u) = c(x_1+y_1, x_1+2) = (cx_1+cy_1, cx_1+2)$ Homogeneity is satisfied.

Thus, $T$ is linear.

Do you want further details on any of the steps? Here are some related questions:

1. How to compute the matrix representation of linear transformations?
2. What is the geometric meaning of the kernel and image of a transformation?
3. How to use rank-nullity theorem in higher dimensions?
4. Can a transformation have both full rank and a non-trivial kernel?
5. What is the practical significance of verifying Sylvester's law?

Tip: Always verify both kernel and image spaces when analyzing linear transformations to ensure consistency