Let's analyze each map step by step. --- ### **(a) \( F : \mathbb{R}^3 \to \mathbb{R}^2 \) given by \( F(x, y, z) = (x, y) \)** #### **Step 1: Check if it's a linear transformation** - **Additivity:** \( F((x_1, y_1, z_1) + (x_2, y_2, z_2)) = F(x_1 + x_2, y_1 + y_2, z_1 + z_2) = (x_1 + x_2, y_1 + y_2) \) \( F(x_1, y_1, z_1) + F(x_2, y_2, z_2) = (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2) \) - **Homogeneity:** \( F(\alpha(x, y, z)) = F(\alpha x, \alpha y, \alpha z) = (\alpha x, \alpha y) = \alpha (x, y) = \alpha F(x, y, z) \) **Conclusion:** Yes, \( F \) is linear. #### **Step 2: Find kernel and image** - **Kernel:** \( F(x, y, z) = (, ) \implies x = , y = \) So, \( \ker F = \{(, , z) \mid z \in \mathbb{R}\} \) - **Image:** Any \((x, y) \in \mathbb{R}^2\) can be written as \( F(x, y, ) \). So, \( \mathrm{Im} F = \mathbb{R}^2 \) --- ### **(b) \( F : \mathbb{R}^4 \to \mathbb{R}^4 \) given by \( F(\mathbf{v}) = -\mathbf{v} \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F(\mathbf{u} + \mathbf{v}) = -(\mathbf{u} + \mathbf{v}) = -\mathbf{u} - \mathbf{v} = F(\mathbf{u}) + F(\mathbf{v}) \) - **Homogeneity:** \( F(\alpha \mathbf{v}) = -(\alpha \mathbf{v}) = \alpha (-\mathbf{v}) = \alpha F(\mathbf{v}) \) **Conclusion:** Yes, \( F \) is linear. #### **Step 2: Kernel and image** - **Kernel:** \( F(\mathbf{v}) = \mathbf{} \implies -\mathbf{v} = \mathbf{} \implies \mathbf{v} = \mathbf{} \). So, \( \ker F = \{ \mathbf{} \} \) - **Image:** For any \( \mathbf{w} \in \mathbb{R}^4 \), \( F(-\mathbf{w}) = \mathbf{w} \). So, \( \mathrm{Im} F = \mathbb{R}^4 \) --- ### **(c) \( F : \mathbb{R}^3 \to \mathbb{R}^3 \) given by \( F(\mathbf{v}) = \mathbf{v} + (, -1, ) \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F(\mathbf{u} + \mathbf{v}) = \mathbf{u} + \mathbf{v} + (, -1, ) \neq (\mathbf{u} + (, -1, )) + (\mathbf{v} + (, -1, )) \) (since the constant is added twice) - **Homogeneity:** \( F(, , ) = (, -1, ) \neq \) **Conclusion:** Not linear. --- ### **(d) \( F : \mathbb{R}^2 \to \mathbb{R}^2 \) given by \( F(x, y) = (2x + y, y) \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F((x_1, y_1) + (x_2, y_2)) = (2(x_1 + x_2) + (y_1 + y_2), y_1 + y_2) \) \( = (2x_1 + 2x_2 + y_1 + y_2, y_1 + y_2) \) \( F(x_1, y_1) + F(x_2, y_2) = (2x_1 + y_1, y_1) + (2x_2 + y_2, y_2) = (2x_1 + y_1 + 2x_2 + y_2, y_1 + y_2) \) They are equal. - **Homogeneity:** \( F(\alpha x, \alpha y) = (2\alpha x + \alpha y, \alpha y) = \alpha(2x + y, y) = \alpha F(x, y) \) **Conclusion:** Yes, \( F \) is linear. #### **Step 2: Kernel and image** - **Kernel:** \( F(x, y) = (, ) \implies 2x + y = , y = \implies x = , y = \) So, \( \ker F = \{(, )\} \) - **Image:** For any \((a, b) \in \mathbb{R}^2\), let \( y = b \), \( x = \frac{a-b}{2} \): \( F\left(\frac{a-b}{2}, b\right) = (a, b) \) So, \( \mathrm{Im} F = \mathbb{R}^2 \) --- ### **(e) \( F : \mathbb{R}^2 \to \mathbb{R}^2 \) given by \( F(x, y) = (y, x) \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F((x_1, y_1) + (x_2, y_2)) = (y_1 + y_2, x_1 + x_2) \) \( F(x_1, y_1) + F(x_2, y_2) = (y_1, x_1) + (y_2, x_2) = (y_1 + y_2, x_1 + x_2) \) - **Homogeneity:** \( F(\alpha x, \alpha y) = (\alpha y, \alpha x) = \alpha (y, x) = \alpha F(x, y) \) **Conclusion:** Yes, \( F \) is linear. #### **Step 2: Kernel and image** - **Kernel:** \( F(x, y) = (, ) \implies y = , x = \implies (x, y) = (, ) \) So, \( \ker F = \{(, )\} \) - **Image:** For any \((a, b) \in \mathbb{R}^2\), \( F(b, a) = (a, b) \), so \( \mathrm{Im} F = \mathbb{R}^2 \) --- ### **(f) \( F : \mathbb{R}^2 \to \mathbb{R} \) given by \( F(x, y) = xy \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F((x_1, y_1) + (x_2, y_2)) = (x_1 + x_2)(y_1 + y_2) = x_1y_1 + x_1y_2 + x_2y_1 + x_2y_2 \) \( F(x_1, y_1) + F(x_2, y_2) = x_1y_1 + x_2y_2 \) Not equal in general. **Conclusion:** Not linear. --- ## **Final Answers** The following are **linear transformations** (with their kernels and images): | Map | Linear? | Kernel | Image | |-----|---------|--------|-------| | a | Yes | \( \{(,,z) | z \in \mathbb{R} \} \) | \( \mathbb{R}^2 \) | | b | Yes | \( \{ \mathbf{} \} \) | \( \mathbb{R}^4 \) | | c | No | - | - | | d | Yes | \( \{(,)\} \) | \( \mathbb{R}^2 \) | | e | Yes | \( \{(,)\} \) | \( \mathbb{R}^2 \) | | f | No | - | - |

Let's analyze each map step by step, determining whether they are linear transformations, and if so, identifying their kernels and images. --- ### **(a) \( F : \mathbb{R}^3 \to \mathbb{R}^2 \) given by \( F(x, y, z) = (2, 9) \)** #### **Step 1: Check if it's a linear transformation** - **Additivity:** \( F((x_1, y_1, z_1) + (x_2, y_2, z_2)) = F(x_1 + x_2, y_1 + y_2, z_1 + z_2) = (2, 9) \) \( F(x_1, y_1, z_1) + F(x_2, y_2, z_2) = (2, 9) + (2, 9) \neq (2, 9) \) - **Homogeneity:** \( F(\alpha(x, y, z)) = F(\alpha x, \alpha y, \alpha z) = (2, 9) \neq \alpha (2, 9) \) **Conclusion:** Not a linear transformation. --- ### **(b) \( F : \mathbb{R}^4 \to \mathbb{R}^4 \) given by \( F(\mathbf{v}) = -\mathbf{v} \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F(\mathbf{u} + \mathbf{v}) = -(\mathbf{u} + \mathbf{v}) = -\mathbf{u} - \mathbf{v} = F(\mathbf{u}) + F(\mathbf{v}) \) - **Homogeneity:** \( F(\alpha \mathbf{v}) = -(\alpha \mathbf{v}) = \alpha (-\mathbf{v}) = \alpha F(\mathbf{v}) \) **Conclusion:** Yes, \( F \) is linear. #### **Step 2: Find kernel and image** - **Kernel:** \( F(\mathbf{v}) = \mathbf{0} \implies -\mathbf{v} = \mathbf{0} \implies \mathbf{v} = \mathbf{0} \). So, \( \ker F = \{ \mathbf{0} \} \) - **Image:** Every vector \( \mathbf{w} \in \mathbb{R}^4 \) can be expressed as \( F(-\mathbf{w}) = \mathbf{w} \). So, \( \mathrm{Im} F = \mathbb{R}^4 \) --- ### **(c) \( F : \mathbb{R}^3 \to \mathbb{R}^3 \) given by \( F(\mathbf{v}) = \mathbf{v} + (0, -1, 0) \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F(\mathbf{u} + \mathbf{v}) = (\mathbf{u} + \mathbf{v}) + (0, -1, 0) \neq F(\mathbf{u}) + F(\mathbf{v}) \) - **Homogeneity:** \( F(\alpha \mathbf{v}) = \alpha \mathbf{v} + (0, -1, 0) \neq \alpha F(\mathbf{v}) \) **Conclusion:** Not a linear transformation. --- ### **(d) \( F : \mathbb{R}^2 \to \mathbb{R}^2 \) given by \( F(x, y) = (2x + y, y) \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F((x_1, y_1) + (x_2, y_2)) = (2(x_1 + x_2) + (y_1 + y_2), y_1 + y_2) = (2x_1 + y_1 + 2x_2 + y_2, y_1 + y_2) \) \( F(x_1, y_1) + F(x_2, y_2) = (2x_1 + y_1, y_1) + (2x_2 + y_2, y_2) = (2x_1 + y_1 + 2x_2 + y_2, y_1 + y_2) \). They are equal. - **Homogeneity:** \( F(\alpha x, \alpha y) = (2\alpha x + \alpha y, \alpha y) = \alpha(2x + y, y) = \alpha F(x, y) \) **Conclusion:** Yes, \( F \) is linear. #### **Step 2: Find kernel and image** - **Kernel:** \( F(x, y) = (0, 0) \implies 2x + y = 0, y = 0 \). Thus, \( x = 0 \) leads to \( \ker F = \{(0, 0)\} \). - **Image:** Any \((a, b) \in \mathbb{R}^2\) can be achieved by choosing \( y = b \) and \( x = \frac{a - b}{2} \), hence \( \mathrm{Im} F = \mathbb{R}^2 \). --- ### **(e) \( F : \mathbb{R}^2 \to \mathbb{R}^2 \) given by \( F(x, y) = (y, x) \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F((x_1, y_1) + (x_2, y_2)) = (y_1 + y_2, x_1 + x_2) = F(x_1, y_1) + F(x_2, y_2) \) - **Homogeneity:** \( F(\alpha x, \alpha y) = (\alpha y, \alpha x) = \alpha (y, x) = \alpha F(x, y) \) **Conclusion:** Yes, \( F \) is linear. #### **Step 2: Find kernel and image** - **Kernel:** \( F(x, y) = (0, 0) \implies y = 0, x = 0 \). Thus, \( \ker F = \{(0, 0)\} \). - **Image:** For any \((a, b) \in \mathbb{R}^2\), choosing \( (b, a) \) yields \( F(b, a) = (a, b) \), hence \( \mathrm{Im} F = \mathbb{R}^2 \). --- ### **(f) \( F : \mathbb{R}^2 \to \mathbb{R} \) given by \( F(x, y) = xy \)** #### **Step 1: Check if it's linear** - **Additivity:** \( F((x_1, y_1) + (x_2, y_2)) = (x_1 + x_2)(y_1 + y_2) = x_1y_1 + x_1y_2 + x_2y_1 + x_2y_2 \) This is not equal to \( F(x_1, y_1) + F(x_2, y_2) \). **Conclusion:** Not a linear transformation. --- ## **Final Answers** The following are **linear transformations** (with their kernels and images): | Map | Linear? | Kernel | Image | |-----|---------|--------|-------| | a | No | - | - | | b | Yes | \( \{ \mathbf{0} \} \) | \( \mathbb{R}^4 \) | | c | No | - | - | | d | Yes | \( \{(0, 0)\} \) | \( \mathbb{R}^2 \) | | e | Yes | \( \{(0, 0)\} \) | \( \mathbb{R}^2 \) | | f | No | - | - |