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To find the matrix \( A \) of the orthogonal projection onto the subspace \( W \) spanned by the vectors 
\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \quad \text{and} \quad \mathbf{v}_2 = \begin{bmatrix} -3 \\ -7 \\ 1 \end{bmatrix},
\]
we can use the formula for the projection matrix onto a subspace spanned by multiple vectors.

### Steps:

1. **Form the matrix of basis vectors**: Arrange \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) as columns in a matrix:
   \[
   V = \begin{bmatrix} 1 & -3 \\ 1 & -7 \\ 1 & 1 \end{bmatrix}.
   \]

2. **Compute \( V^T V \)**: Find the product of \( V \) transposed and \( V \) itself.

   \[
   V^T V = \begin{bmatrix} 1 & 1 & 1 \\ -3 & -7 & 1 \end{bmatrix} \begin{bmatrix} 1 & -3 \\ 1 & -7 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 3 & -9 \\ -9 & 59 \end{bmatrix}.
   \]

3. **Compute \( (V^T V)^{-1} \)**: Invert \( V^T V \).
   \[
   (V^T V)^{-1} = \begin{bmatrix} 3 & -9 \\ -9 & 59 \end{bmatrix}^{-1}.
   \]
   Calculating the inverse gives:
   \[
   (V^T V)^{-1} = \begin{bmatrix} \frac{59}{32} & \frac{9}{32} \\ \frac{9}{32} & \frac{3}{32} \end{bmatrix}.
   \]

4. **Compute \( A = V (V^T V)^{-1} V^T \)**: Finally, calculate the projection matrix.

   Substitute values to get \( A \):
   \[
   A = \begin{bmatrix} 1 & -3 \\ 1 & -7 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} \frac{59}{32} & \frac{9}{32} \\ \frac{9}{32} & \frac{3}{32} \end{bmatrix} \begin{bmatrix} 1 & 1 & 1 \\ -3 & -7 & 1 \end{bmatrix}.
   \]

After performing the matrix multiplication, you’ll obtain the matrix \( A \), which represents the orthogonal projection onto \( W \).

Would you like further clarification on any steps, or do you want to see the full calculation of matrix multiplications?

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Here are five related questions to expand understanding:

1. How does one compute the orthogonal projection matrix if the subspace is spanned by a single vector?
2. What are the properties of an orthogonal projection matrix?
3. How does changing one of the basis vectors affect the projection matrix?
4. How can you verify that a matrix is indeed a projection matrix?
5. What role does the Gram-Schmidt process play in constructing projection matrices?

**Tip:** When computing matrix inverses by hand, remember that it's often simpler to work with row operations than determinants for larger matrices.

Let W be the subspace of R^3 spanned by the vectors [1, 1, 1] and [-3, -7, 1]. Find the matrix A of the orthogonal projection onto W.

Explore the solution to find the orthogonal projection matrix onto a subspace W in R^3, spanned by vectors [1, 1, 1] and [-3, -7, 1]. Learn how to use matrix multiplication and the projection formula to compute the matrix A of the projection. Suitable for linear algebra students at the undergraduate level.

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