Math Problem Statement
Solution
To find a basis for the row space of matrix , we need to identify the linearly independent rows of . The row space of a matrix is spanned by its linearly independent rows.
Given:
Steps to find a basis for the row space:
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Row Reduce Matrix : Perform row operations to bring the matrix to row echelon form or reduced row echelon form (RREF). The non-zero rows in RREF will be linearly independent and thus form a basis for the row space.
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Identify Linearly Independent Rows: After row reduction, select the rows that contain pivot elements (leading 1s) as the basis vectors for the row space.
Let's proceed with the row reduction.
After performing Gaussian elimination, the RREF of would look like:
Basis for Row Space: The non-zero rows in this RREF matrix are:
Thus, a basis for the row space of is:
You can input this basis as:
1 0 2 0
0 1 -1 0
0 0 0 1
Would you like further details on the row reduction process or have questions?
Here are 5 related questions to deepen understanding:
- What is the significance of a basis in linear algebra?
- Why do we only consider linearly independent rows when finding the row space basis?
- How does row reduction help in identifying linearly independent vectors?
- What are pivot elements, and why are they essential in basis determination?
- Can the row space basis change if we permute the rows of the matrix before reduction?
Tip: A basis for a row space is unique in terms of the span, but the specific vectors can vary based on row operations, as any set of linearly independent rows that span the row space will form a valid basis.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Row Space
Basis of a Matrix
Gaussian Elimination
Formulas
Row Echelon Form (REF)
Reduced Row Echelon Form (RREF)
Theorems
Basis Theorem
Row Space and Linear Independence
Suitable Grade Level
College Level