Math Problem Statement

Find the orthogonal projection of v⃗ =⎡⎣⎢⎢⎢13,−10,8,−9⎤⎦⎥⎥⎥ onto the subspace W of R4 spanned by ⎡⎣⎢⎢⎢321−4⎤⎦⎥⎥⎥ and ⎡⎣⎢⎢⎢−2,−5,16,0⎤⎦⎥⎥⎥. projW(v⃗ )= ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢ ⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

Solution

To find the orthogonal projection of v\mathbf{v} onto the subspace WW, we use the following steps:

  1. Let the basis of WW be given by w1\mathbf{w}_1 and w2\mathbf{w}_2, where: w1=[3214],w2=[25160].\mathbf{w}_1 = \begin{bmatrix} 3 \\ 2 \\ 1 \\ -4 \end{bmatrix}, \quad \mathbf{w}_2 = \begin{bmatrix} -2 \\ -5 \\ 16 \\ 0 \end{bmatrix}.

    The orthogonal projection of v\mathbf{v} onto WW is: projW(v)=c1w1+c2w2,\text{proj}_W(\mathbf{v}) = c_1 \mathbf{w}_1 + c_2 \mathbf{w}_2, where c1c_1 and c2c_2 are scalars to be determined.

  2. To determine c1c_1 and c2c_2, we solve: c1=vw1w1w1,c2=vw2w2w2.c_1 = \frac{\mathbf{v} \cdot \mathbf{w}_1}{\mathbf{w}_1 \cdot \mathbf{w}_1}, \quad c_2 = \frac{\mathbf{v} \cdot \mathbf{w}_2}{\mathbf{w}_2 \cdot \mathbf{w}_2}.

  3. Compute the necessary dot products:

    • v=[131089]\mathbf{v} = \begin{bmatrix} 13 \\ -10 \\ 8 \\ -9 \end{bmatrix},
    • w1=[3214]\mathbf{w}_1 = \begin{bmatrix} 3 \\ 2 \\ 1 \\ -4 \end{bmatrix},
    • w2=[25160]\mathbf{w}_2 = \begin{bmatrix} -2 \\ -5 \\ 16 \\ 0 \end{bmatrix}.

Let’s compute these values.import numpy as np

Define the vectors

v = np.array([13, -10, 8, -9]) w1 = np.array([3, 2, 1, -4]) w2 = np.array([-2, -5, 16, 0])

Compute dot products and norms

v_dot_w1 = np.dot(v, w1) w1_dot_w1 = np.dot(w1, w1)

v_dot_w2 = np.dot(v, w2) w2_dot_w2 = np.dot(w2, w2)

Compute the coefficients

c1 = v_dot_w1 / w1_dot_w1 c2 = v_dot_w2 / w2_dot_w2

Compute the projection

proj_w1 = c1 * w1 proj_w2 = c2 * w2 proj_v = proj_w1 + proj_w2

proj_varray([ 5.23333333, 1.53333333, 10.63333333, -8.4 ])array([ 5.23333333, 1.53333333, 10.63333333, -8.4 ])successThe orthogonal projection of v=[131089]\mathbf{v} = \begin{bmatrix} 13 \\ -10 \\ 8 \\ -9 \end{bmatrix} onto the subspace WW spanned by w1=[3214]\mathbf{w}_1 = \begin{bmatrix} 3 \\ 2 \\ 1 \\ -4 \end{bmatrix} and w2=[25160]\mathbf{w}_2 = \begin{bmatrix} -2 \\ -5 \\ 16 \\ 0 \end{bmatrix} is:

projW(v)=[5.2331.53310.6338.4].\text{proj}_W(\mathbf{v}) = \begin{bmatrix} 5.233 \\ 1.533 \\ 10.633 \\ -8.4 \end{bmatrix}.

Would you like a detailed breakdown of the calculations? Let me know if you have further questions.

Here are 5 related questions:

  1. How do you compute projections onto higher-dimensional subspaces?
  2. What is the geometric interpretation of the orthogonal projection in Rn\mathbb{R}^n?
  3. How does Gram-Schmidt relate to orthogonal projections?
  4. Can you explain how to verify if vprojW(v)\mathbf{v} - \text{proj}_W(\mathbf{v}) is orthogonal to WW?
  5. What happens if the spanning vectors are not linearly independent?

Tip: Always ensure your spanning vectors are linearly independent for a subspace basis.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Orthogonal Projection

Formulas

Projection formula: proj_W(v) = c1 * w1 + c2 * w2
Coefficients: c1 = (v ⋅ w1) / (w1 ⋅ w1), c2 = (v ⋅ w2) / (w2 ⋅ w2)

Theorems

Orthogonal Projection Theorem
Properties of Dot Product

Suitable Grade Level

College Level (Linear Algebra)