Find the matrix corresponding the orthogonal projection in R2
 onto the subspace spanned by the vector

To find the matrix corresponding to the orthogonal projection onto the subspace spanned by a vector **v** in \( \mathbb{R}^2 \), we proceed as follows:

Given a vector \( \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \), the projection of any vector \( \mathbf{u} = \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \) onto \( \mathbf{v} \) is given by:

\[
\text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}
\]

### Step-by-Step:

1. **Find the dot product** \( \mathbf{v} \cdot \mathbf{v} \):
   \[
   \mathbf{v} \cdot \mathbf{v} = v_1^2 + v_2^2
   \]

2. **Form the projection formula**:
   \[
   \text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{u_1 v_1 + u_2 v_2}{v_1^2 + v_2^2} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}
   \]

3. **Matrix representation**:
   The matrix \( P \) that represents the projection is the matrix such that \( P\mathbf{u} = \text{proj}_{\mathbf{v}}(\mathbf{u}) \).

   For a projection onto \( \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \), the matrix \( P \) is given by:

   \[
   P = \frac{1}{v_1^2 + v_2^2} \begin{pmatrix} v_1^2 & v_1 v_2 \\ v_1 v_2 & v_2^2 \end{pmatrix}
   \]

### Final Matrix Formula:
\[
P = \frac{1}{v_1^2 + v_2^2} \begin{pmatrix} v_1^2 & v_1 v_2 \\ v_1 v_2 & v_2^2 \end{pmatrix}
\]

This matrix \( P \) projects any vector in \( \mathbb{R}^2 \) onto the subspace spanned by the vector \( \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \).

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Would you like to go over an example or clarify any part of this process? Here are five questions to consider:

1. How do you verify that the matrix \( P \) is a projection matrix?
2. How do you check that \( P \) is an orthogonal projection matrix?
3. What is the difference between orthogonal and oblique projections?
4. How does the projection matrix behave for special cases like \( \mathbf{v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \)?
5. Can this method be generalized to projections in \( \mathbb{R}^n \)?

**Tip:** The matrix of an orthogonal projection is always symmetric and idempotent (i.e., \( P^2 = P \)).

Find the matrix corresponding to the orthogonal projection in R2 onto the subspace spanned by the vector.

Learn how to find the matrix corresponding to the orthogonal projection onto a subspace in R2 spanned by a vector. This solution covers key concepts in linear algebra, including projection formulas and matrix representation. Suitable for college-level students studying linear algebra.

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