Given that $A: \mathbb{R}^3 \to \mathbb{R}^2$ and we have the basis vectors $\{v_1, v_2, v_3\}$ for $\mathbb{R}^3$, with the set $\{A v_1, A v_2\}$ being linearly independent in $\mathbb{R}^2$, the question asks about the number of solutions to the equation $A(x) = 0$, where $x \in \mathbb{R}^3$.

Step-by-Step Analysis:

1. Linear Transformation Properties:

   • $A$ is a linear transformation from a 3-dimensional space ( $\mathbb{R}^3$ ) to a 2-dimensional space ( $\mathbb{R}^2$ ).
   • The matrix representing $A$ must have dimensions $2 \times 3$, since it's mapping from $\mathbb{R}^3$ to $\mathbb{R}^2$.

2. Rank of the Matrix $A$:

   • We are given that $\{A v_1, A v_2\}$ is a linearly independent set in $\mathbb{R}^2$. This implies that the image of $A$ contains at least two linearly independent vectors in $\mathbb{R}^2$.
   • Therefore, the rank of $A$, which corresponds to the number of linearly independent columns, is at least 2.
   • Since $A$ is a $2 \times 3$ matrix, the maximum possible rank of $A$ is 2 (because the number of rows in $A$ is 2).

3. Dimension of the Kernel (Null Space):

   • By the rank-nullity theorem, we have: $\text{dim}(\text{Ker}(A)) + \text{dim}(\text{Im}(A)) = 3$
   • The rank of $A$ is 2, so: $\text{dim}(\text{Ker}(A)) + 2 = 3$ $\text{dim}(\text{Ker}(A)) = 1$

4. Conclusion:

   • The dimension of the kernel (null space) of $A$ is 1. This means that the null space of $A$ is a 1-dimensional subspace of $\mathbb{R}^3$, implying there is a nontrivial solution to $A(x) = 0$.
   • Since the null space is 1-dimensional, there are infinitely many solutions to the equation $A(x) = 0$, because every scalar multiple of a nonzero vector in the kernel is also a solution.

Final Answer:

The equation $A(x) = 0$ has infinitely many solutions, as the null space of $A$ has dimension 1, meaning there is a 1-dimensional space of solutions.

Would you like further details on the rank-nullity theorem or kernel and image concepts?

Here are 5 related questions for deeper understanding:

1. How does the rank-nullity theorem apply to transformations between spaces of different dimensions?
2. What are the implications of a transformation having a rank less than the number of columns?
3. Can the kernel of a matrix ever be trivial (just the zero vector)?
4. How do the concepts of linear independence and rank relate to each other?
5. What conditions must hold for a transformation to be injective?

Tip: When analyzing transformations, always check the dimensions of both the domain and codomain to predict the behavior of solutions, especially in solving equations like $A(x) = 0$.