Math Problem Statement
Solution
To find an orthogonal basis for the column space of the given matrix, we can use the Gram-Schmidt orthogonalization process on the column vectors of the matrix. The matrix is:
-1 & 5 & 5 \\ 3 & -6 & 3 \\ 2 & -1 & 5 \\ 1 & -5 & -4 \\ \end{bmatrix}$$ Let’s denote the columns as $$\mathbf{v}_1 = \begin{bmatrix} -1 \\ 3 \\ 2 \\ 1 \end{bmatrix}$$, $$\mathbf{v}_2 = \begin{bmatrix} 5 \\ -6 \\ -1 \\ -5 \end{bmatrix}$$, and $$\mathbf{v}_3 = \begin{bmatrix} 5 \\ 3 \\ 5 \\ -4 \end{bmatrix}$$. ### Step 1: Apply Gram-Schmidt to Find Orthogonal Vectors 1. **Set** $$\mathbf{u}_1 = \mathbf{v}_1$$. 2. **Find** $$\mathbf{u}_2$$ by making $$\mathbf{v}_2$$ orthogonal to $$\mathbf{u}_1$$: $$\mathbf{u}_2 = \mathbf{v}_2 - \frac{\mathbf{v}_2 \cdot \mathbf{u}_1}{\mathbf{u}_1 \cdot \mathbf{u}_1} \mathbf{u}_1$$ 3. **Find** $$\mathbf{u}_3$$ by making $$\mathbf{v}_3$$ orthogonal to both $$\mathbf{u}_1$$ and $$\mathbf{u}_2$$: $$\mathbf{u}_3 = \mathbf{v}_3 - \frac{\mathbf{v}_3 \cdot \mathbf{u}_1}{\mathbf{u}_1 \cdot \mathbf{u}_1} \mathbf{u}_1 - \frac{\mathbf{v}_3 \cdot \mathbf{u}_2}{\mathbf{u}_2 \cdot \mathbf{u}_2} \mathbf{u}_2$$ ### Step 2: Normalize the Vectors (if an orthonormal basis is needed) After completing these steps, the vectors $$\mathbf{u}_1$$, $$\mathbf{u}_2$$, and $$\mathbf{u}_3$$ (if all are linearly independent) will form an orthogonal basis for the column space of the matrix. Let me calculate these for you.import numpy as np # Define the matrix and extract its columns matrix = np.array([ [-1, 5, 5], [3, -6, 3], [2, -1, 5], [1, -5, -4] ]) # Extract columns as vectors v1 = matrix[:, 0] v2 = matrix[:, 1] v3 = matrix[:, 2] # Gram-Schmidt process def gram_schmidt(vectors): basis = [] for v in vectors: w = v - sum(np.dot(v, b) / np.dot(b, b) * b for b in basis) if np.linalg.norm(w) > 1e-10: # Avoid adding zero vectors basis.append(w) return basis # Apply Gram-Schmidt to the columns of the matrix orthogonal_basis = gram_schmidt([v1, v2, v3]) # Show the result as a list of orthogonal vectors orthogonal_basis[array([-1, 3, 2, 1]), array([ 3., 0., 3., -3.]), array([ 1., 1., -1., 0.])][array([-1, 3, 2, 1]), array([ 3., 0., 3., -3.]), array([ 1., 1., -1., 0.])]successThe orthogonal basis for the column space of the given matrix is: $$\left\{ \begin{bmatrix} -1 \\ 3 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 3 \\ -3 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ -1 \\ 0 \end{bmatrix} \right\}$$ If you need further details on each step of the process or have any questions, feel free to ask! Here are some follow-up questions you might find helpful: 1. How can I find an orthonormal basis instead of an orthogonal basis? 2. Why does the Gram-Schmidt process work for finding orthogonal vectors? 3. Can I use other methods to find a basis for the column space? 4. What happens if the matrix columns are linearly dependent? 5. How can I verify that the basis vectors are orthogonal? **Tip:** To simplify calculations, especially for larger matrices, consider using software like Python or MATLAB to perform Gram-Schmidt orthogonalization.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Column Space
Orthogonal Basis
Gram-Schmidt Orthogonalization
Formulas
Orthogonalization formula for vector u_2: u_2 = v_2 - (v_2 • u_1 / u_1 • u_1) * u_1
Orthogonalization formula for vector u_3: u_3 = v_3 - (v_3 • u_1 / u_1 • u_1) * u_1 - (v_3 • u_2 / u_2 • u_2) * u_2
Theorems
Gram-Schmidt Process
Suitable Grade Level
Undergraduate Level
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