Math Problem Statement
(d) Find a basis for the column space of A.
Find a basis for the column space of A.
Solution
To find a basis for the column space of matrix , we first need to determine which columns are linearly independent. Since the matrices and are given to be row-equivalent, both have the same rank, which is 3. This means that exactly 3 columns of are linearly independent, and these columns will form a basis for the column space of .
Here’s the step-by-step approach to determine the basis:
1. Identify Pivot Columns from
Matrix is in row echelon form:
\begin{pmatrix} 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & -2 & 0 & 3 \\ 0 & 0 & 1 & 0 & -5 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$$ From the non-zero rows of $$B$$, the **pivot columns** (the ones with the leading 1 in each row) are the **1st, 2nd, and 3rd columns**. These columns correspond to the independent columns in matrix $$A$$. ### 2. Extract the Basis Columns from $$A$$ Now, we take the **1st, 2nd, and 3rd columns** from matrix $$A$$: $$A = \begin{pmatrix} -2 & -5 & 8 & 0 & -17 \\ 1 & 3 & -5 & 1 & 5 \\ 1 & 5 & -9 & 5 & -9 \\ 1 & 7 & -13 & 5 & -3 \end{pmatrix}$$ So, the first three columns of $$A$$ are: $$\text{Column 1:} \quad \begin{pmatrix} -2 \\ 1 \\ 1 \\ 1 \end{pmatrix}, \quad \text{Column 2:} \quad \begin{pmatrix} -5 \\ 3 \\ 5 \\ 7 \end{pmatrix}, \quad \text{Column 3:} \quad \begin{pmatrix} 8 \\ -5 \\ -9 \\ -13 \end{pmatrix}$$ ### 3. Write the Basis for the Column Space The set of these three columns forms a basis for the **column space of $$A$$**: $$\left\{ \begin{pmatrix} -2 \\ 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -5 \\ 3 \\ 5 \\ 7 \end{pmatrix}, \begin{pmatrix} 8 \\ -5 \\ -9 \\ -13 \end{pmatrix} \right\}$$ ### Final Answer The **basis for the column space of $$A$$** is: $$\left\{ \begin{pmatrix} -2 \\ 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -5 \\ 3 \\ 5 \\ 7 \end{pmatrix}, \begin{pmatrix} 8 \\ -5 \\ -9 \\ -13 \end{pmatrix} \right\}$$ This is a set of three linearly independent vectors in $$\mathbb{R}^4$$, which spans the column space of $$A$$. --- Do you have any questions or need further clarifications? Here are 5 related questions: 1. How can you determine the rank of a matrix without row reduction? 2. What is the dimension of the null space of $$A$$ given its nullity? 3. How do row equivalence and column equivalence differ? 4. How would you find a basis for the **null space** of $$A$$? 5. Can two matrices with the same rank have different column spaces? **Tip:** A basis is not unique; different sets of independent columns can form valid bases for the same column space.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Column Space
Basis
Rank
Nullity
Formulas
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Theorems
Rank-Nullity Theorem
Suitable Grade Level
Undergraduate level (College Linear Algebra)