Let's analyze the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$ given that it maps every vector in $\mathbb{R}^2$ to a vector on the $x_1$-axis in $\mathbb{R}^3$.

Key Information:

• $T$ maps every vector in $\mathbb{R}^2$ to the $x_1$-axis in $\mathbb{R}^3$.
  • This means the image (or range) of any vector $\mathbf{x} \in \mathbb{R}^2$ will be of the form $\mathbf{T}(\mathbf{x}) = [a, 0, 0]^T$, where $a \in \mathbb{R}$.
  • Hence, all vectors are mapped to the $x_1$-axis, which is 1-dimensional.

Matrix Representation $A$:

• The standard matrix $A$ of the transformation $T$ is a $3 \times 2$ matrix (since $T$ maps from $\mathbb{R}^2$ to $\mathbb{R}^3$).
• Since all vectors in $\mathbb{R}^2$ are mapped to the $x_1$-axis in $\mathbb{R}^3$, the first row of $A$ will have non-zero values, while the second and third rows will consist of zeros, indicating no movement in the $x_2$ or $x_3$ directions.

Evaluating the Statements:

1. $T(\mathbf{x}) = \mathbf{0}$ has only the trivial solution $\mathbf{x} = \mathbf{0}$:

   • Since the mapping takes all vectors to the $x_1$-axis, this is not necessarily true. There could be non-trivial solutions where $\mathbf{x} \neq \mathbf{0}$ but $T(\mathbf{x}) = \mathbf{0}$.

2. Range of $T$ is $\mathbb{R}^3$:

   • False, because the range of $T$ is confined to the $x_1$-axis, which is a 1-dimensional subspace of $\mathbb{R}^3$, not all of $\mathbb{R}^3$.

3. $T$ is onto:

   • False, because for $T$ to be onto, the range must cover all of $\mathbb{R}^3$, which is not the case here (the range is only the $x_1$-axis).

4. The RREF of $A$ has at least two rows of zeros:

   • True. Since $T$ maps all vectors in $\mathbb{R}^2$ to the $x_1$-axis, the matrix $A$ will have zeros in the second and third rows. Thus, in reduced row echelon form (RREF), there will be at least two rows of zeros.

Correct Answer:

• The RREF of $A$ has at least two rows of zeros.

---

Would you like to see more details on how we arrive at this conclusion, or do you have any other questions?

Here are 5 related questions to expand on this topic:

1. How do you find the standard matrix for a linear transformation?
2. What is the difference between "onto" and "one-to-one" for linear transformations?
3. How does the rank of a matrix relate to the dimensionality of its image?
4. What are the properties of a transformation that maps vectors to a line in $\mathbb{R}^3$?
5. How do you compute the RREF of a matrix?

Tip: For linear transformations, always check the rank of the matrix to determine the dimension of the image (range).