Determine if the following linear mapping is injective, surjective, bijective or none of these.

L:R2→R2 defined by L(x1,x2)=(2x1+2x2,3x1)

Injective

Surjective

Bijective

None of the above

To determine if the linear mapping \( L: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( L(x_1, x_2) = (2x_1 + 2x_2, 3x_1) \) is injective, surjective, bijective, or none, let’s analyze it step-by-step.

### Step 1: Represent \( L \) as a Matrix
The mapping \( L(x_1, x_2) = (2x_1 + 2x_2, 3x_1) \) can be expressed as a matrix:
\[
L(x_1, x_2) = \begin{pmatrix} 2 & 2 \\ 3 & 0 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}.
\]
This matrix representation is:
\[
A = \begin{pmatrix} 2 & 2 \\ 3 & 0 \end{pmatrix}.
\]

### Step 2: Determine Injectivity (One-to-One)
To check if \( L \) is injective, we need to see if the kernel of \( L \) (the set of vectors mapped to \( 0 \)) only contains the zero vector. For injectivity, the matrix \( A \) must have a pivot in every column, meaning it must be of full column rank.

1. Calculate the determinant of \( A \):
   \[
   \det(A) = (2)(0) - (2)(3) = -6.
   \]
Since \( \det(A) \neq 0 \), \( A \) is invertible, meaning that the mapping is injective.

### Step 3: Determine Surjectivity (Onto)
Because \( A \) is a \( 2 \times 2 \) matrix with a non-zero determinant, it spans all of \( \mathbb{R}^2 \). Therefore, the image of \( L \) covers the entire codomain \( \mathbb{R}^2 \), meaning that \( L \) is also surjective.

### Conclusion: Bijectivity
Since \( L \) is both injective and surjective, it is **bijective**.

**Answer:** The mapping \( L \) is **bijective**.

Would you like further details or have any questions?

---

1. How is the determinant of a matrix used to determine bijectivity?
2. What conditions make a linear mapping non-injective?
3. How does one determine the rank of a matrix in general?
4. Why does a non-zero determinant imply a full rank?
5. What would happen if we changed one entry of matrix \( A \)?

**Tip:** For linear transformations, injectivity and surjectivity are determined by examining the matrix’s rank and its determinant in the case of square matrices.

Determine if the following linear mapping is injective, surjective, bijective or none of these. L:R2→R2 defined by L(x1,x2)=(2x1+2x2,3x1)

Explore the analysis of a linear mapping L: R2 → R2 defined by L(x1, x2) = (2x1 + 2x2, 3x1). This comprehensive guide covers the concepts of injectivity, surjectivity, and bijectivity with detailed steps and explanations. Ideal for students in Grades 11-12.

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