Math Problem Statement

Find a basis of the given subspace by deleting linearly dependent vectors from the span of three vectors.

Solution

To find a basis for the given subspace, we need to examine the three vectors provided and remove any that are linearly dependent. The three vectors given are:

v1=[0001],v2=[50−10],v3=[3033]v_1 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 5 \\ 0 \\ -1 \\ 0 \end{bmatrix}, \quad v_3 = \begin{bmatrix} 3 \\ 0 \\ 3 \\ 3 \end{bmatrix}

The method to find the basis is as follows:

Step 1: Form a matrix from the vectors.

Construct a matrix AA using the given vectors as its columns:

0 & 5 & 3 \\ 0 & 0 & 0 \\ 0 & -1 & 3 \\ 1 & 0 & 3 \end{bmatrix}$$ ### Step 2: Perform row reduction. We will perform Gaussian elimination (row reduction) on the matrix $$A$$ to determine if the vectors are linearly independent. If a row of all zeros appears during the reduction, it indicates linear dependence. The row-reduced form of $$A$$ (also called the row echelon form) is: $$\text{Row-reduced form of } A = \begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ ### Step 3: Identify linearly independent vectors. From the row-reduced form, we can see that the first two columns contain pivots, meaning they correspond to linearly independent vectors. The third column is a linear combination of the first two, so it is dependent. ### Step 4: Basis for the subspace. The linearly independent vectors are $$v_1 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$$ and $$v_2 = \begin{bmatrix} 5 \\ 0 \\ -1 \\ 0 \end{bmatrix}$$. Thus, a basis for the subspace is: $$\left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 5 \\ 0 \\ -1 \\ 0 \end{bmatrix} \right\}$$ Let me know if you would like more details or clarifications! --- Here are 5 related questions that could deepen your understanding: 1. What is the general method to determine linear independence of vectors? 2. How can you apply Gaussian elimination to find the rank of a matrix? 3. Why is the concept of a "basis" important in vector spaces? 4. How do you interpret the dimension of a subspace based on its basis? 5. Can you explain how vector spaces are related to systems of linear equations? ### Tip: When checking for linear dependence, remember that a set of vectors is dependent if one vector can be written as a linear combination of the others.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Linear Independence

Formulas

Gaussian elimination to find the row-reduced echelon form
Condition for linear dependence (Ax = 0)

Theorems

Basis theorem
Linear independence criterion

Suitable Grade Level

Undergraduate

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