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For the given transformation T decide if it is linear. If so, construct the matrix representation of T in the standard bases of the vector spaces involved.

Let's analyze the linearity of the given transformations \( T \) and, if they are linear, construct the matrix representations in the standard bases of the vector spaces involved.

### Part (a): Transformation \( T: P^2 \to P^3 \)
This transformation is defined by:
\[
T(p(x)) = xp(x) + 1
\]
where \( p(x) \) is a polynomial of degree at most 2.

#### Linearity Check:
For \( T \) to be linear, it must satisfy the following properties:
1. **Additivity:** \( T(p(x) + q(x)) = T(p(x)) + T(q(x)) \)
2. **Homogeneity:** \( T(c \cdot p(x)) = c \cdot T(p(x)) \) for any scalar \( c \).

**1. Additivity:**
Let's consider two polynomials \( p(x) \) and \( q(x) \).
\[
T(p(x) + q(x)) = x(p(x) + q(x)) + 1 = xp(x) + xq(x) + 1
\]
\[
T(p(x)) + T(q(x)) = (xp(x) + 1) + (xq(x) + 1) = xp(x) + xq(x) + 2
\]
Since \( T(p(x) + q(x)) \neq T(p(x)) + T(q(x)) \), **\( T \) is not additive** and therefore not linear.

Since \( T \) is not linear, we do not construct a matrix representation for it.

### Part (b): Transformation \( T: \mathbb{R}^3 \to \mathbb{R}^2 \)
This transformation is defined by:
\[
T\left(\begin{pmatrix} x \\ y \\ z \end{pmatrix}\right) = \begin{pmatrix} 2x + y \\ 3y - 4z \end{pmatrix}
\]

#### Linearity Check:
Again, we check additivity and homogeneity.

**1. Additivity:**
For vectors \( \mathbf{u} = \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix} \) and \( \mathbf{v} = \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix} \):
\[
T(\mathbf{u} + \mathbf{v}) = T\left(\begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \\ z_1 + z_2 \end{pmatrix}\right) = \begin{pmatrix} 2(x_1 + x_2) + (y_1 + y_2) \\ 3(y_1 + y_2) - 4(z_1 + z_2) \end{pmatrix}
\]
\[
= \begin{pmatrix} 2x_1 + 2x_2 + y_1 + y_2 \\ 3y_1 + 3y_2 - 4z_1 - 4z_2 \end{pmatrix} = \begin{pmatrix} 2x_1 + y_1 \\ 3y_1 - 4z_1 \end{pmatrix} + \begin{pmatrix} 2x_2 + y_2 \\ 3y_2 - 4z_2 \end{pmatrix} = T(\mathbf{u}) + T(\mathbf{v})
\]

**2. Homogeneity:**
For a scalar \( c \):
\[
T(c \mathbf{u}) = T\left(\begin{pmatrix} c x \\ c y \\ c z \end{pmatrix}\right) = \begin{pmatrix} 2c x + c y \\ 3c y - 4c z \end{pmatrix} = c \begin{pmatrix} 2x + y \\ 3y - 4z \end{pmatrix} = c T(\mathbf{u})
\]

Since both conditions are satisfied, \( T \) is a **linear transformation**.

#### Matrix Representation of \( T \):
To find the matrix representation of \( T \), we apply \( T \) to the standard basis vectors of \( \mathbb{R}^3 \):
- \( T\left(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\right) = \begin{pmatrix} 2(1) + 0 \\ 3(0) - 4(0) \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix} \)
- \( T\left(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\right) = \begin{pmatrix} 2(0) + 1(1) \\ 3(1) - 4(0) \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \end{pmatrix} \)
- \( T\left(\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\right) = \begin{pmatrix} 2(0) + 0 \\ 3(0) - 4(1) \end{pmatrix} = \begin{pmatrix} 0 \\ -4 \end{pmatrix} \)

Thus, the matrix representation of \( T \) is:
\[
[T] = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & -4 \end{pmatrix}
\]

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Would you like any further details on these transformations or have any additional questions?

### Related Questions:
1. What are the properties that define a linear transformation?
2. How do you determine if a given transformation is bijective?
3. What is the relationship between the matrix of a linear transformation and its effect on vector spaces?
4. How do you find the inverse of a linear transformation?
5. Can you determine the image (range) and kernel (null space) of a linear transformation from its matrix?

**Tip:** When determining the linearity of a transformation, always check both additivity and homogeneity.

Explore how to determine the linearity of given transformations and construct matrix representations in the standard bases of vector spaces involved. Includes detailed examples and solutions.

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