To determine whether the mapping $F : \mathbb{R}^3 \to \mathbb{R}^2$, defined by

$F(x_1, x_2, x_3) = (9x_1 - 8x_2, 6x_1 - 9x_3),$

is a linear map, we need to check if it satisfies the two conditions for linearity:

1. Additivity: $F(\mathbf{u} + \mathbf{v}) = F(\mathbf{u}) + F(\mathbf{v})$ for any vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^3$.
2. Homogeneity: $F(c \mathbf{u}) = c F(\mathbf{u})$ for any scalar $c \in \mathbb{R}$ and vector $\mathbf{u} \in \mathbb{R}^3$.

Since the mapping $F$ is given in terms of linear combinations of $x_1$, $x_2$, and $x_3$, we can check if each output component is a linear combination of the input components without any constant terms.

Step-by-Step Check

The components of $F(x_1, x_2, x_3)$ are:

• The first component is $9x_1 - 8x_2$.
• The second component is $6x_1 - 9x_3$.

Both components are linear functions of $x_1$, $x_2$, and $x_3$, and there are no constant (non-variable) terms. This implies that both additivity and homogeneity hold for this mapping.

Therefore, the map $F$ is linear.

Answer:

• The correct choice is: The map $F$ is linear.

Would you like more details on the properties of linear maps or have any questions?

Here are some follow-up questions to explore this topic further:

1. What are the general criteria for determining if a mapping between vector spaces is linear?
2. How would the presence of constant terms in the mapping affect its linearity?
3. Can a mapping be linear if it includes squared terms or other nonlinear functions of the inputs?
4. What would the matrix representation of this linear map $F$ be?
5. How does linearity of a map impact transformations in geometry?

Tip: When determining if a map is linear, always check for additivity and homogeneity, and ensure there are no constant terms or nonlinear components in the mapping.