Suppose T is a linear transformation from R ^ 2 to p_{2} such that RBT: U CO:CO1 Marks: 8 ii) 7 * [[1], [1]] = 2 - 3x + x ^ 2 and r * [[2], [3]] = 1 - x ^ 2 T * [[1], [- 2]] r * [[a], [b]]

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:
\[
\begin{aligned}
1 &= a + 2b \\
-2 &= a + 3b
\end{aligned}
\]

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.

Suppose T is a linear transformation from R^2 to p_{2} such that RBT: U CO:CO1 Marks: 8 ii) 7 * [[1], [1]] = 2 - 3x + x^2 and r * [[2], [3]] = 1 - x^2 T * [[1], [-2]] r * [[a], [b]]

Explore a linear transformation problem from R^2 to P2 where T([1,-2]^T) is computed using given images of vectors under the transformation. The solution involves solving a system of linear equations and applying the linearity of transformations. Suitable for undergraduate students learning Linear Algebra.

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