How to find a basis of a matrix?
Understand the Problem
The question is asking for a method to determine a basis of a matrix, which typically involves finding the column space or the row space of the matrix and identifying a set of linearly independent vectors that span these spaces.
Answer
The basis consists of the original columns corresponding to the pivot columns in the Row Echelon Form.
Answer for screen readers
The basis of the given matrix consists of the original columns that correspond to the pivot columns identified in the Row Echelon Form.
Steps to Solve
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Write the Matrix in Row Echelon Form (REF) To find a basis, you first need to convert the matrix into Row Echelon Form (REF). Use Gaussian elimination to perform row operations until you reach the REF. This means that each leading entry (the first non-zero number from the left, in a row) is 1, and all entries below the leading entry are zeros.
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Identify the Pivot Columns After obtaining the REF, identify the pivot columns. The pivot columns are those that contain the leading 1s in each row. This helps to locate the columns in the original matrix that correspond to the leading 1s.
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Extract the Basis Vectors Take the columns from the original matrix corresponding to the pivot columns. These columns form the basis for the column space of the matrix. If you are interested in the row space, take the non-zero rows from the REF.
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Check for Linear Independence To make sure that the selected set of vectors are linearly independent, you can form a matrix with these columns and check the rank. If the rank equals the number of vectors, they are linearly independent.
The basis of the given matrix consists of the original columns that correspond to the pivot columns identified in the Row Echelon Form.
More Information
Finding a basis of a matrix helps in understanding its structure, such as its column space and row space. This is essential in applications such as solving systems of linear equations and understanding the span of vector spaces.
Tips
- Failing to accurately perform row operations leading to incorrect REF.
- Not identifying all pivot columns correctly.
- Assuming that all columns in the REF are part of the basis instead of just the pivot columns.
