Let's break this down step by step. You're dealing with a linear transformation $T$ from $\mathbb{R}^2$ to $P_2$, where $P_2$ is the space of polynomials of degree at most 2. The transformation acts on vectors in $\mathbb{R}^2$, and the images of specific vectors under $T$ are given as polynomials in $P_2$.

Key Information Given:

1. $T( [1, 1]^T ) = 2 - 3x + x^2$
2. $T( [2, 3]^T ) = 1 - x^2$
3. Find $T( [1, -2]^T )$.

1. Linear Transformation Definition:

Since $T$ is a linear transformation, we know that for any vectors $u, v \in \mathbb{R}^2$ and scalars $a, b$, the transformation satisfies:

$T(a \mathbf{u} + b \mathbf{v}) = a T(\mathbf{u}) + b T(\mathbf{v})$

We'll use this property to find $T( [1, -2]^T )$.

2. Construct the Solution Using Given Information:

You are given the images of two specific vectors under $T$:

$T( [1, 1]^T ) = 2 - 3x + x^2$ $T( [2, 3]^T ) = 1 - x^2$

Now, we need to express $[1, -2]^T$ as a linear combination of $[1, 1]^T$ and $[2, 3]^T$.

Let's solve the system:

[ [1, -2]^T = a[1, 1]^T + b[2, 3]^T ]

This leads to the equations: $1 = a + 2b$ $-2 = a + 3b$

3. Solve the System of Equations:

We now have the system:

1 &= a + 2b \\ -2 &= a + 3b \end{aligned}$$ Subtract the first equation from the second: $$(-2 - 1) = (a + 3b) - (a + 2b)$$ $$-3 = b$$ Substitute $$b = -3$$ into the first equation: $$1 = a + 2(-3)$$ $$1 = a - 6$$ $$a = 7$$ So, we have: \[ [1, -2]^T = 7[1, 1]^T - 3[2, 3]^T \] ### 4. Apply the Linear Transformation: Using the linearity of $$T$$, we can now compute $$T( [1, -2]^T )$$: \[ T( [1, -2]^T ) = 7T( [1, 1]^T ) - 3T( [2, 3]^T ) \] Substitute the given values for $$T( [1, 1]^T )$$ and $$T( [2, 3]^T )$$: $$T( [1, -2]^T ) = 7(2 - 3x + x^2) - 3(1 - x^2)$$ Now, distribute the constants: $$T( [1, -2]^T ) = 14 - 21x + 7x^2 - 3 + 3x^2$$ Simplify: $$T( [1, -2]^T ) = (14 - 3) + (-21x) + (7x^2 + 3x^2)$$ $$T( [1, -2]^T ) = 11 - 21x + 10x^2$$ ### Final Answer: $$T( [1, -2]^T ) = 11 - 21x + 10x^2$$ --- Would you like more details on any specific step? Here are some related questions to consider: 1. How do you find the matrix representation of a linear transformation? 2. How do linear transformations between vector spaces work in higher dimensions? 3. Can you always express a vector as a linear combination of other vectors? 4. What are some examples of linear transformations from $$\mathbb{R}^2$$ to $$P_2$$? 5. How does linearity simplify the process of finding images of vectors under a transformation? **Tip**: Always check if you can express a vector as a combination of known vectors to simplify the computation of transformations.