Let T: R2 R2 be a linear transform Time left 0:26:15 R2 is a vector space over the field R, defined by T(x1, x2) = ((x1+x2)/2, (x1 + x2)/2). What is the corresponding matrix for T?

Oa. 1/2 1/2 1/2 1/2

Ob. 0 0 1/2 1/2

OC. 1/2 1/2 0 0

Od [1/2 1/2] d. 0

The transformation \( T \) is defined by:

\[
T(x_1, x_2) = \left( \frac{x_1 + x_2}{2}, \frac{x_1 + x_2}{2} \right)
\]

We want to express this linear transformation in matrix form. To do this, we apply \( T \) to the standard basis vectors of \( \mathbb{R}^2 \):

1. Apply \( T \) to \( (1, 0) \):
   \[
   T(1, 0) = \left( \frac{1 + 0}{2}, \frac{1 + 0}{2} \right) = \left( \frac{1}{2}, \frac{1}{2} \right)
   \]
2. Apply \( T \) to \( (0, 1) \):
   \[
   T(0, 1) = \left( \frac{0 + 1}{2}, \frac{0 + 1}{2} \right) = \left( \frac{1}{2}, \frac{1}{2} \right)
   \]

These results form the columns of the transformation matrix. So, the matrix \( A \) representing \( T \) is:

\[
A = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}
\]

Thus, the correct option is:

**Oa. \( \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} \)**

Would you like further details on the transformation or how matrices represent linear transformations?

Here are 5 related questions to expand on this:

1. How do we find the matrix of a linear transformation in general?
2. What are the properties of the matrix found for this transformation?
3. How do we verify if a given transformation is linear?
4. How does this transformation affect vectors geometrically in \( \mathbb{R}^2 \)?
5. What would happen if the transformation was not linear—how would that affect the matrix representation?

**Tip**: For any linear transformation, applying it to the standard basis vectors helps derive the corresponding matrix.

Let T: R2 → R2 be a linear transform. R2 is a vector space over the field R, defined by T(x1, x2) = ((x1 + x2) / 2, (x1 + x2) / 2). What is the corresponding matrix for T?

Options:
Oa. 1/2 1/2 1/2 1/2
Ob. 0 0 1/2 1/2
Oc. 1/2 1/2 0 0
Od. [1/2 1/2] 0

Discover how to determine the matrix for the linear transformation T(x1, x2) = ((x1 + x2) / 2, (x1 + x2) / 2) over the vector space R2. This problem involves key concepts in linear algebra, including basis vectors and matrix representation of linear transformations. Ideal for undergraduate mathematics students.

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