Math Problem Statement

Consider the subspace of \( \mathbb{R}^4 \) spanned by the vectors \([-8, 2, 0, 12]^T, [5, -2, -8, -6]^T, [-2, -3, 8, 6]^T\). Create a basis for this subspace and express the vector \([1, 1, 2, -3]^T\) in terms of this basis.

Solution

To solve this problem, we need to determine a basis for the subspace UU in R4\mathbb{R}^4, which is spanned by the given vectors. Then, we express xx in terms of the chosen basis vectors.

Steps to Solve:

  1. 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}.$$
  2. Row reduce the matrix: Use Gaussian elimination to reduce AA to row echelon form. This will allow us to determine the linearly independent vectors (basis vectors).

  3. Extract basis vectors: The columns corresponding to the pivot positions in the row-reduced matrix form the basis for the subspace UU.

  4. Solve for xx: If {v1,v2}\{ v_1, v_2 \} are the basis vectors, write xx as a linear combination of these basis vectors: x=c1v1+c2v2,x = c_1 v_1 + c_2 v_2, where c1c_1 and c2c_2 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 UU in R4\mathbb{R}^4. 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!

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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