91Ó°ÊÓ

In the realm of linear transformations, additivity plays a crucial role. As part of the definition of a linear transformation, additivity ensures that when you apply a transformation to a sum of vectors, it is equivalent to applying the transformation to each vector individually and then adding the results.
For our transformation \( T \) given by\[ T\left[\begin{array}{l} x \ y \ z \end{array}\right]=\left[\begin{array}{c} x \ x+y \ x+y+z \end{array}\right] \]we need to verify additivity.
To do so, take any two vectors \( \mathbf{u} = \begin{bmatrix} x_1 \ y_1 \ z_1 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} x_2 \ y_2 \ z_2 \end{bmatrix} \), and consider their sum \( \mathbf{u} + \mathbf{v} = \begin{bmatrix} x_1 + x_2 \ y_1 + y_2 \ z_1 + z_2 \end{bmatrix} \).
When we look at \( T(\mathbf{u} + \mathbf{v}) \), it translates to- \( T\left(\begin{bmatrix} x_1 + x_2 \ y_1 + y_2 \ z_1 + z_2 \end{bmatrix}\right) = \begin{bmatrix} x_1 + x_2 \ x_1 + x_2 + y_1 + y_2 \ x_1 + x_2 + y_1 + y_2 + z_1 + z_2 \end{bmatrix} \)Then compare this with finding \( T(\mathbf{u}) + T(\mathbf{v}) \):- \( T(\mathbf{u}) = \begin{bmatrix} x_1 \ x_1 + y_1 \ x_1 + y_1 + z_1 \end{bmatrix} \)- \( T(\mathbf{v}) = \begin{bmatrix} x_2 \ x_2 + y_2 \ x_2 + y_2 + z_2 \end{bmatrix} \)- Summing these results gives: \( \begin{bmatrix} x_1 + x_2 \ x_1 + y_1 + x_2 + y_2 \ x_1 + y_1 + z_1 + x_2 + y_2 + z_2 \end{bmatrix} \)The two results are identical, confirming that additivity holds for this transformation.

Homogeneity is another essential characteristic of linear transformations. It ensures that scaling a vector before a transformation produces the same result as transforming a vector and then scaling it. This property holds for linear transformations, and to prove it, we check if \( T(c\mathbf{u}) = c T(\mathbf{u}) \) for any scalar \( c \) and vector \( \mathbf{u} \).
Given our transformation, let's verify homogeneity:
Set \( \mathbf{u} = \begin{bmatrix} x \ y \ z \end{bmatrix} \) and multiply it by a scalar \( c \). This gives us \( c\mathbf{u} = \begin{bmatrix} cx \ cy \ cz \end{bmatrix} \).
After applying the transformation to this scaled vector, we get:- \( T\left(\begin{bmatrix} cx \ cy \ cz \end{bmatrix}\right) = \begin{bmatrix} cx \ cx + cy \ cx + cy + cz \end{bmatrix} \)
On the other hand, when we apply the transformation to \( \mathbf{u} \) first, we have:- \( T(\mathbf{u}) = \begin{bmatrix} x \ x + y \ x + y + z \end{bmatrix} \)
Multiplying this result by \( c \), we obtain:- \( cT(\mathbf{u}) = \begin{bmatrix} cx \ c(x + y) \ c(x + y + z) \end{bmatrix} = \begin{bmatrix} cx \ cx + cy \ cx + cy + cz \end{bmatrix} \)The results from \( T(c\mathbf{u}) \) and \( cT(\mathbf{u}) \) are identical. This confirms that homogeneity is satisfied in this transformation.

Matrix transformations are a fundamental aspect of understanding linear transformations, as they provide a structured way to apply these transformations using matrices. A matrix transformation takes a vector and maps it to another vector through matrix multiplication. In our example, while the transformation is explicitly given, understanding matrix transformations helps conceptualize the process.
A transformation like- \( T\left(\begin{bmatrix} x \ y \ z \end{bmatrix}\right) = \begin{bmatrix} x \ x+y \ x+y+z \end{bmatrix} \)can also be represented as a multiplication with a suitable matrix. Identifying this matrix can help generalize the transformation:- Let \( A \) be the matrix such that \( A \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} x \ x+y \ x+y+z \end{bmatrix} \), then \( A \) takes the form:\[A = \begin{bmatrix}1 & 0 & 0 \1 & 1 & 0 \1 & 1 & 1 \\end{bmatrix}\]When this matrix multiplies a vector \( \begin{bmatrix} x \ y \ z \end{bmatrix} \), it yields the same result as our transformation \( T \). Matrix transformations thus allow us to analyze properties like invertibility, range, and kernel using matrix theory.