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., $P^2 = P$).