Question

Which of the following is/are true ?

• A. Every linear transformation from \(\R^2\) to \(\R^2\) maps lines onto points or lines
• B. Every surjective linear transformation from \(\R^2\) to \(\R^2\) maps lines onto lines
• C. Every bijective linear transformation from \(\R^2\) to \(\R^2\) maps pairs of parallel lines to pairs of parallel lines
• D. Every bijective linear transformation from \(\R^2\) to \(\R^2\) maps pairs of perpendicular lines to pairs of perpendicular lines

Answer 1

The Correct Option is A, B, C

Let's analyze each statement one by one to determine which ones are true: 

1. Every linear transformation from \((\R^2)\) to \((\R^2)\) maps lines onto points or lines: 
   A linear transformation from \((\R^2)\) to \((\R^2)\) can be represented by a \( 2 \times 2 \) matrix \( A \). For a line in \(\R^2\) described by \( \mathbf{x} = \mathbf{a} + t\mathbf{b} \), where \( t \) is a scalar, its image under \( A \) will be \( A\mathbf{x} = A\mathbf{a} + tA\mathbf{b} \). Depending on \( A \mathbf{b} \), this can either:
   • Be a point if \( A\mathbf{b} = \mathbf{0} \) (i.e., \( \mathbf{b} \) is in the null space of \( A \)),
   • Or remain a line if \( A\mathbf{b} \neq \mathbf{0} \).
2. Every surjective linear transformation from \((\R^2)\) to \((\R^2)\) maps lines onto lines: 
   A surjective transformation means that the rank of the transformation matrix is 2, implying that it maps \((\R^2)\) onto the whole of \((\R^2)\) without losing dimensions. Thus, lines remain as lines under such transformations, as there is no dimension reduction to a point. Therefore, the statement is true.
3. Every bijective linear transformation from \((\R^2)\) to \((\R^2)\) maps pairs of parallel lines to pairs of parallel lines: 
   A bijective transformation is both injective and surjective, meaning the transformation is a one-to-one correspondence and covers the entire codomain. The parallelism of lines is preserved under bijective (invertible) linear transformations because these transformations maintain vector space properties, including direction and scaling. Thus, this statement is true.
4. Every bijective linear transformation from \((\R^2)\) to \((\R^2)\) maps pairs of perpendicular lines to pairs of perpendicular lines: 
   While bijective transformations preserve linearity and parallelism, they do not necessarily preserve angles. Therefore, pairs of perpendicular lines may not remain perpendicular after such a transformation, especially if rotation or scaling alters angles. Thus, this statement is false.

Therefore, the correct statements are:

• Every linear transformation from \((\R^2)\) to \((\R^2)\) maps lines onto points or lines.
• Every surjective linear transformation from \((\R^2)\) to \((\R^2)\) maps lines onto lines.
• Every bijective linear transformation from \((\R^2)\) to \((\R^2)\) maps pairs of parallel lines to pairs of parallel lines.