Math Problem Statement

Question content area top Part 1 Let Bold v 1equalsleft bracket Start 3 By 1 Matrix 1st Row 1st Column 0 2nd Row 1st Column 0 3rd Row 1st Column negative 2 EndMatrix right bracket ​, Bold v 2equalsleft bracket Start 3 By 1 Matrix 1st Row 1st Column 0 2nd Row 1st Column negative 4 3rd Row 1st Column negative 4 EndMatrix right bracket ​, and Bold v 3equalsleft bracket Start 3 By 1 Matrix 1st Row 1st Column 6 2nd Row 1st Column negative 5 3rd Row 1st Column negative 6 EndMatrix right bracket . Does StartSet Bold v 1 comma Bold v 2 comma Bold v 3 EndSet span set of real numbers R cubed​? Why or why​ not?

Solution

To determine whether the set {v1,v2,v3}\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} spans R3\mathbb{R}^3, we need to check if these vectors are linearly independent and if they form a basis for R3\mathbb{R}^3.

The vectors are:

v1=[00−2],v2=[0−4−4],v3=[6−5−6]\mathbf{v}_1 = \begin{bmatrix} 0 \\ 0 \\ -2 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0 \\ -4 \\ -4 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 6 \\ -5 \\ -6 \end{bmatrix}

Step 1: Form the matrix

We construct a matrix by placing v1\mathbf{v}_1, v2\mathbf{v}_2, and v3\mathbf{v}_3 as the columns of a 3x3 matrix:

0 & 0 & 6 \\ 0 & -4 & -5 \\ -2 & -4 & -6 \end{bmatrix}$$ ### Step 2: Row reduce the matrix We need to row-reduce this matrix to determine if the vectors are linearly independent. If the matrix has full rank (rank 3), then the vectors span $$\mathbb{R}^3$$. 1. The first column is already simplified since the first two entries are zero. 2. The third row is already in reduced form for the first column. 3. Now, work on reducing the second and third columns. I will perform these calculations and row-reduce the matrix to check the rank.The row-reduced form of the matrix is the identity matrix: $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ This indicates that the matrix has full rank (rank 3), meaning that the vectors $$\mathbf{v}_1$$, $$\mathbf{v}_2$$, and $$\mathbf{v}_3$$ are linearly independent. ### Conclusion: Since the vectors are linearly independent, they span $$\mathbb{R}^3$$. Therefore, the set $$\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}$$ **does** span $$\mathbb{R}^3$$. --- Would you like more details or have any further questions? Here are five related questions to consider: 1. What is the geometric interpretation of spanning $$\mathbb{R}^3$$? 2. How can you check if two vectors span a plane in $$\mathbb{R}^3$$? 3. Can a set of dependent vectors ever span $$\mathbb{R}^3$$? 4. How does the rank of a matrix relate to the number of vectors that span a space? 5. What happens if one of the vectors is the zero vector in terms of spanning? **Tip:** Linear independence is crucial for determining the span of vectors in a space—always check this property by analyzing the matrix's rank.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Linear Independence
Matrix Row Reduction
Span

Formulas

Matrix Row Reduction
Rank of a Matrix

Theorems

Linear Independence Criterion
Span of a Set of Vectors

Suitable Grade Level

Undergraduate

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