Let’s break down the questions and address them step by step.

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

The transformation $T$ takes $(1, 2)$ to $(3, -1, 5)$ and $(3, 1)$ to $(3, 1, -1)$.
We are tasked to find $T(-1, 1)$.

Solution:

The transformation is linear, so $T(a, b) = a \cdot T(1, 2) + b \cdot T(3, 1)$.
Let’s compute $T(-1, 1)$:
$T(-1, 1) = (-1) \cdot (3, -1, 5) + (1) \cdot (3, 1, -1)$
$= (-3, 1, -5) + (3, 1, -1) = (0, 2, -6)$
Thus, $T(-1, 1) = (0, 2, -6)$.

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2. Transformation $P$: $\mathbb{R}^3 \to \mathbb{R}^2$

Defined as $P(x, y, z) = (x + z, y + z)$.

We are asked to find the kernel, image, nullity, and rank of $P$.

Kernel of $P$:

The kernel is the set of all vectors $(x, y, z) \in \mathbb{R}^3$ such that $P(x, y, z) = (0, 0)$.
From $P(x, y, z) = (x + z, y + z)$:
$x + z = 0 \quad \text{and} \quad y + z = 0$
This implies $x = -z$ and $y = -z$, so $(x, y, z) = (-z, -z, z) = z(-1, -1, 1)$.
The kernel is therefore:
$\ker(P) = \text{span}\{(-1, -1, 1)\}.$

Image of $P$:

The image is the set of all vectors $(a, b) \in \mathbb{R}^2$ that can be written as $P(x, y, z) = (x + z, y + z)$.
Setting $a = x + z$ and $b = y + z$, we see that any $(a, b)$ is possible. Thus:
$\text{Im}(P) = \mathbb{R}^2.$

Nullity of $P$:

The nullity is the dimension of the kernel:
$\text{nullity}(P) = 1.$

Rank of $P$:

The rank is the dimension of the image:
$\text{rank}(P) = 2.$

From the Rank-Nullity Theorem, we confirm:
$\text{rank}(P) + \text{nullity}(P) = 3.$

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3. Verification of Sylvester’s Law of Inertia:

Sylvester’s law states that the number of positive, negative, and zero eigenvalues of a symmetric matrix remains invariant under congruence transformations.

In this case, verifying Sylvester's law depends on the quadratic form represented by the transformation $P(x, y, z)$. Let me know if you want detailed eigenvalue verification for this.

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If you'd like me to explore any part of this in greater depth, feel free to ask!

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Questions to Expand Your Understanding:

1. What are the geometric implications of the kernel and image of a linear transformation?
2. How does the rank-nullity theorem help classify linear transformations?
3. Can you provide another example of a transformation to apply Sylvester’s law?
4. How do we compute eigenvalues for a transformation matrix to verify its properties?
5. Can you explore the relationship between the dimensions of the domain and codomain of $P$?

Tip:

Always use the Rank-Nullity Theorem to double-check your kernel and image computations!