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 onto the subspace , we use the following steps:
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Let the basis of be given by and , where:
The orthogonal projection of onto is: where and are scalars to be determined.
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To determine and , we solve:
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Compute the necessary dot products:
- ,
- ,
- .
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 onto the subspace spanned by and is:
Would you like a detailed breakdown of the calculations? Let me know if you have further questions.
Here are 5 related questions:
- How do you compute projections onto higher-dimensional subspaces?
- What is the geometric interpretation of the orthogonal projection in ?
- How does Gram-Schmidt relate to orthogonal projections?
- Can you explain how to verify if is orthogonal to ?
- 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)
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