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.