Number 40 show all work ### Linear Equations in Linear Algebra**34.** Let \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) be the transformation that reflects each vector \( \mathbf{x} = (x_1, x_2, x_3) \) through the plane \( x_3 = 0 \) onto \( T(\mathbf{x}) = (x_1, x_2, -x_3) \). Show that \( T \) is a linear transformation. [See Example 4 for ideas.]**35.** Let \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) be the transformation that projects each vector \( \mathbf{x} = (x_1, x_2, x_3) \) onto the plane \( x_2 = 0 \), so \( T(\mathbf{x}) = (x_1, 0, x_3) \). Show that \( T \) is a linear transformation.**36.** Let \( T : \mathbb{R}^n \to \mathbb{R}^m \) be a linear transformation. Suppose \( \{ \mathbf{u}, \mathbf{v} \} \) is a linearly independent set, but \( \{ T(\mathbf{u}), T(\mathbf{v}) \} \) is a linearly dependent set. Show that \( T(\mathbf{x}) = 0 \) has a nontrivial solution.[**Hint:** Use the fact that \( c_1 T(\mathbf{u}) + c_2 T(\mathbf{v}) = 0 \) for some weights \( c_1 \) and \( c_2 \), not both zero.] **[M]** In Exercises 37 and 38, the given matrix determines a linear transformation \( T \). Find all \( \mathbf{x} \) such that \( T(\mathbf{x}) = \mathbf{0} \).**37.** \[ \begin{bmatrix} 2 & 3 & 5 \\ -7 & 3 & -4 \\ -9 & 3 & 6 \end{bmatrix}\]**38.** \[ \begin{bmatrix} 3 & -

Name: Number 40 show all work ### Linear Equations in Linear Algebra**34.**
Uploaded: 2024-03-05T11:52:21.000Z
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Description: Number 40 show all work ### Linear Equations in Linear Algebra**34.** Let \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) be the transformation that reflects each vector \( \mathbf{x} = (x_1, x_2, x_3) \) through the plane \( x_3 = 0 \) onto \( T(\mathbf{x})