Questions: Find the orthogonal projection of v = [ 15, -14, 8, -20 ] onto the subspace V of R^4 spanned by x1 = [ -3, -2, 1, -1 ] and x2 = [ 2, 5, 16, 0 ]

Transcript text: Find the orthogonal projection of \[ \mathbf{v}=\left[\begin{array}{c} 15 \\ -14 \\ 8 \\ -20 \end{array}\right] \] onto the subspace $V$ of $\mathbb{R}^{4}$ spanned by \[ \mathbf{x}_{1}=\left[\begin{array}{c} -3 \\ -2 \\ 1 \\ -1 \end{array}\right] \text { and } \mathbf{x}_{2}=\left[\begin{array}{c} 2 \\ 5 \\ 16 \\ 0 \end{array}\right] \]

Solution

Solution Steps

To find the orthogonal projection of a vector $\mathbf{v}$ onto a subspace $V$ spanned by vectors $\mathbf{x}_1$ and $\mathbf{x}_2$, we can use the following steps:

Form the matrix $A$ whose columns are the basis vectors $\mathbf{x}_1$ and $\mathbf{x}_2$.
Compute the projection matrix $P = A(A^T A)^{-1} A^T$.
Multiply the projection matrix $P$ by the vector $\mathbf{v}$ to get the orthogonal projection.

Step 1: Form the Matrix $A$

Given the basis vectors $\mathbf{x}_1$ and $\mathbf{x}_2$: \[ \mathbf{x}_1 = \begin{bmatrix} -3 \\ -2 \\ 1 \\ -1 \end{bmatrix}, \quad \mathbf{x}_2 = \begin{bmatrix} 2 \\ 5 \\ 16 \\ 0 \end{bmatrix} \] We form the matrix $A$ by using these vectors as columns: \[ A = \begin{bmatrix} -3 & 2 \\ -2 & 5 \\ 1 & 16 \\ -1 & 0 \end{bmatrix} \]

Step 2: Compute the Projection Matrix $P$

The projection matrix $P$ is given by: \[ P = A (A^T A)^{-1} A^T \] First, compute $A^T A$: \[ A^T A = \begin{bmatrix} -3 & -2 & 1 & -1 \\ 2 & 5 & 16 & 0 \end{bmatrix} \begin{bmatrix} -3 & 2 \\ -2 & 5 \\ 1 & 16 \\ -1 & 0 \end{bmatrix} = \begin{bmatrix} 15 & -1 \\ -1 & 285 \end{bmatrix} \] Next, compute $(A^T A)^{-1}$: \[ (A^T A)^{-1} = \begin{bmatrix} 15 & -1 \\ -1 & 285 \end{bmatrix}^{-1} = \begin{bmatrix} \frac{285}{4284} & \frac{1}{4284} \\ \frac{1}{4284} & \frac{15}{4284} \end{bmatrix} \] Now, compute $P$: \[ P = A \begin{bmatrix} \frac{285}{4284} & \frac{1}{4284} \\ \frac{1}{4284} & \frac{15}{4284} \end{bmatrix} A^T = \begin{bmatrix} 0.6140 & 0.4351 & -0.0877 & 0.2 \\ 0.4351 & 0.3544 & 0.1474 & 0.1333 \\ -0.0877 & 0.1474 & 0.9649 & -0.0667 \\ 0.2 & 0.1333 & -0.0667 & 0.0667 \end{bmatrix} \]

Step 3: Compute the Orthogonal Projection

Given the vector $\mathbf{v}$: \[ \mathbf{v} = \begin{bmatrix} 15 \\ -14 \\ 8 \\ -20 \end{bmatrix} \] The orthogonal projection of $\mathbf{v}$ onto the subspace $V$ is: \[ \text{projection} = P \mathbf{v} = \begin{bmatrix} 0.6140 & 0.4351 & -0.0877 & 0.2 \\ 0.4351 & 0.3544 & 0.1474 & 0.1333 \\ -0.0877 & 0.1474 & 0.9649 & -0.0667 \\ 0.2 & 0.1333 & -0.0667 & 0.0667 \end{bmatrix} \begin{bmatrix} 15 \\ -14 \\ 8 \\ -20 \end{bmatrix} = \begin{bmatrix} -1.5825 \\ 0.0772 \\ 5.6737 \\ -0.7333 \end{bmatrix} \]

Final Answer

The orthogonal projection of $\mathbf{v}$ onto the subspace $V$ is: \[ \boxed{\begin{bmatrix} -1.5825 \\ 0.0772 \\ 5.6737 \\ -0.7333 \end{bmatrix}} \]

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