A linear transformation T : R2 ! R3 maps every vectors in R2 to a vector on
the x1-axis in R3. Let A be the standard matrix of T. Which of the following statements
is true? Select only one.
T(~x) = ~0 has only the trivial solution ~x = ~0
 Range of T is R.
T is onto.
The RREF of A has at least two rows of zeros

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**.

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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).

A linear transformation T : R2 -> R3 maps every vector in R2 to a vector on the x1-axis in R3. Let A be the standard matrix of T. Which of the following statements is true? Select only one.
- T(~x) = ~0 has only the trivial solution ~x = ~0
- Range of T is R.
- T is onto.
- The RREF of A has at least two rows of zeros

Explore a linear transformation T: R2 -> R3 that maps vectors to the x1-axis. Learn why the RREF of the matrix A has two rows of zeros and why T is not onto. This problem explains key concepts such as linear transformations, matrix representation, and vector spaces. Suitable for undergraduate students.

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