Math Problem Statement
Solution
To solve this problem, we need to determine a basis for the subspace in , which is spanned by the given vectors. Then, we express in terms of the chosen basis vectors.
Steps to Solve:
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Construct the matrix: Arrange the given vectors as columns of a matrix:
-8 & 5 & -2 \\ 2 & -2 & -3 \\ 0 & -8 & 8 \\ 12 & -6 & 6 \end{bmatrix}.$$ -
Row reduce the matrix: Use Gaussian elimination to reduce to row echelon form. This will allow us to determine the linearly independent vectors (basis vectors).
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Extract basis vectors: The columns corresponding to the pivot positions in the row-reduced matrix form the basis for the subspace .
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Solve for : If are the basis vectors, write as a linear combination of these basis vectors: where and are constants to be determined.
Let me calculate this.import numpy as np from sympy import Matrix
Original matrix with columns as given vectors
A = Matrix([ [-8, 5, -2], [2, -2, -3], [0, -8, 8], [12, -6, 6] ])
Perform row reduction
rref_matrix, pivot_columns = A.rref()
Extract basis vectors
basis_vectors = [A[:, i] for i in pivot_columns]
Display results for the user
basis_vectors, rref_matrix([Matrix([ [-8], [ 2], [ 0], [12]]), Matrix([ [ 5], [-2], [-8], [-6]]), Matrix([ [-2], [-3], [ 8], [ 6]])], Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1], [0, 0, 0]]))([Matrix([ [-8], [ 2], [ 0], [12]]), Matrix([ [ 5], [-2], [-8], [-6]]), Matrix([ [-2], [-3], [ 8], [ 6]])], Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1], [0, 0, 0]]))successThe row reduction process reveals that all three original vectors are linearly independent. Thus, they all form a basis for the subspace in . The basis vectors are:
-8 \\ 2 \\ 0 \\ 12 \end{bmatrix}, \begin{bmatrix} 5 \\ -2 \\ -8 \\ -6 \end{bmatrix}, \begin{bmatrix} -2 \\ -3 \\ 8 \\ 6 \end{bmatrix}.$$ Next, we need to express $$x$$ in terms of this basis. Let me calculate $$x$$.# Vector x provided in the problem x = Matrix([1, 1, 2, -3]) # Solve for coefficients c1, c2, c3 such that x = c1*v1 + c2*v2 + c3*v3 basis_matrix = Matrix.hstack(*basis_vectors) coefficients = basis_matrix.LUsolve(x) # Display coefficients coefficientsMatrix([ [-3/8], [-1/2], [-1/4]])Matrix([ [-3/8], [-1/2], [-1/4]])successThe vector $$x = \begin{bmatrix} 1 \\ 1 \\ 2 \\ -3 \end{bmatrix}$$ can be expressed as a linear combination of the basis vectors as follows: $$x = \frac{-3}{8} \begin{bmatrix} -8 \\ 2 \\ 0 \\ 12 \end{bmatrix} + \frac{-1}{2} \begin{bmatrix} 5 \\ -2 \\ -8 \\ -6 \end{bmatrix} + \frac{-1}{4} \begin{bmatrix} -2 \\ -3 \\ 8 \\ 6 \end{bmatrix}.$$ So the coefficients are: $$c_1 = -\frac{3}{8}, \quad c_2 = -\frac{1}{2}, \quad c_3 = -\frac{1}{4}.$$ Would you like more details or have any other questions? Here are some related questions to explore: 1. What are the properties of the span of a set of vectors in $$\mathbb{R}^n$$? 2. How does row reduction help identify linearly independent vectors? 3. What is the significance of the coefficients in the linear combination? 4. Can this method be extended to other dimensions or spaces? 5. What happens if the given vectors are not linearly independent? **Tip**: Always verify linear independence when forming a basis for any vector space!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Subspace
Span
Basis
Linear Independence
Formulas
Gaussian elimination for row reduction
Linear combination of vectors
Theorems
Basis and dimension theorem
Span and linear independence
Suitable Grade Level
College-level Linear Algebra