Math Problem Statement

Consider the subspace U = span{ [-6, 2, -2, -2], [-3, 3, 9, -3], [-9, -5, -3, -5] } of R^4. Create a basis for U.

Solution

To determine a basis for $U$ , we will analyze the given vectors and determine their linear independence. Let us process the problem step by step:

Problem Overview

Given Vectors: The subspace $U$ is defined as the span of the set of vectors in $\mathbb{R}^4$ :
$\mathbf{v_2} = \begin{bmatrix} -3 \\ 3 \\ 9 \\ -3 \end{bmatrix}, \quad \mathbf{v_3} = \begin{bmatrix} -9 \\ -5 \\ -3 \\ -5 \end{bmatrix}.$$$
Goal: Find a linearly independent set of vectors from the span that forms a basis for $U$ .
Procedure:
- Construct a matrix $A$ with the vectors $\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}$ as columns.
- Perform row-reduction (Gaussian elimination) on $A$ to find the pivot columns.
- The corresponding vectors to the pivot columns will form the basis.

Let me calculate this systematically.import numpy as np from sympy import Matrix

Define the vectors as columns of a matrix

A = np.array([ [-6, -3, -9], [ 2, 3, -5], [-2, 9, -3], [-2, -3, -5] ])

Perform row-reduction (RREF) using sympy

A_matrix = Matrix(A) A_rref, pivots = A_matrix.rref()

Output the results of row-reduction

A_rref, pivots(Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1], [0, 0, 0]]), (0, 1, 2))(Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1], [0, 0, 0]]), (0, 1, 2))successThe row-reduced echelon form (RREF) of the matrix $A$ is:

1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}.$$ The pivot columns are columns $$0, 1,$$ and $$2$$. Therefore, the given vectors $$\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}$$ are linearly independent, and they themselves form the basis for $$U$$. ### Basis for $$U$$: $$\left\{ \begin{bmatrix} -6 \\ 2 \\ -2 \\ -2 \end{bmatrix}, \begin{bmatrix} -3 \\ 3 \\ 9 \\ -3 \end{bmatrix}, \begin{bmatrix} -9 \\ -5 \\ -3 \\ -5 \end{bmatrix} \right\}.$$ If you have more questions or need further details, feel free to ask! --- 1. Would you like the explicit computation of the RREF process? 2. How can we verify linear independence more intuitively? 3. What is the geometric interpretation of the basis in $$\mathbb{R}^4$$? 4. How can we test if additional vectors are in this subspace? 5. Would you like to explore the dimension of $$U$$ and its implications? **Tip**: When finding a basis, always ensure that the vectors you select are linearly independent, which can be systematically checked using RREF!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Basis of a Subspace
Linear Independence

Formulas

Matrix Row Reduction
Conditions for Linear Independence

Theorems

Theorem: A basis of a subspace is a linearly independent set that spans the subspace.
RREF (Row-Reduced Echelon Form)

Suitable Grade Level

Undergraduate (Linear Algebra Course)

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