Math Problem Statement

Find the basis of a column space and write linear combination

Solution

To find the basis of a column space of a matrix and express each column as a linear combination of the basis vectors, follow these steps:

Step 1: Identify the Matrix

Let’s denote the matrix as AA. For illustration, assume:

1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix}$$ ### Step 2: Perform Gaussian Elimination Convert the matrix to its row echelon form (REF) or reduced row echelon form (RREF) by performing row operations. **RREF of $$A$$:** $$R = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ ### Step 3: Identify the Pivot Columns The columns corresponding to the pivot positions (non-zero rows in the REF) form the basis for the column space. In the matrix $$R$$, the pivot is in the first column. ### Step 4: Extract the Basis Vectors The basis for the column space of $$A$$ is the set of original columns in $$A$$ corresponding to the pivot columns in $$R$$. Here, the first column of $$A$$ is the only pivot column. Thus, the basis for the column space of $$A$$ is: $$\text{Basis} = \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \right\}$$ ### Step 5: Write Linear Combinations Each column in the original matrix $$A$$ can be expressed as a linear combination of the basis vectors: - First column: $$\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = 1 \times \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$ - Second column: $$\begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix} = 2 \times \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$ - Third column: $$\begin{pmatrix} 3 \\ 6 \\ 9 \end{pmatrix} = 3 \times \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$ So, the basis is $$\left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \right\}$$, and each column is a scalar multiple of this basis vector. --- Would you like more details or have any questions? Here are some related questions to consider: 1. What is the dimension of the column space? 2. How do you determine if a set of vectors is linearly independent? 3. Can a matrix have more than one basis for its column space? 4. What is the relationship between the rank of a matrix and its column space? 5. How do you find the null space of a matrix? 6. How do row operations affect the column space? 7. What is the geometric interpretation of the column space? 8. How can you find a basis for the row space of a matrix? **Tip:** Always ensure the set of basis vectors is linearly independent to correctly represent the column space.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Column Space
Linear Combination

Formulas

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Theorems

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Suitable Grade Level

Advanced Undergraduate

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