How to find the basis of the column space?

Understand the Problem

The question is asking for the method to determine the basis of the column space of a matrix, which involves linear algebra concepts like row reduction and identifying linearly independent columns.

Answer

The basis of the column space is formed by the columns of the original matrix corresponding to the pivot columns after row reduction.

Steps to Solve

Set Up the Matrix Write down the matrix for which you want to find the basis of the column space.
Perform Row Reduction Use Gaussian elimination or row-echelon form to simplify the matrix. Aim for a row echelon form so that you can easily identify the pivot columns.
Identify Pivot Columns Look at the row-reduced form of the matrix and determine which columns contain the leading 1s (pivots). These columns are crucial for the next step.
Select Corresponding Columns from Original Matrix Select the columns from the original matrix that correspond to the pivot columns you identified in the previous step.
Express the Basis The set of selected columns from the original matrix forms a basis for the column space.

The basis of the column space is formed by the columns of the original matrix that correspond to the pivot columns in the row-reduced form.

More Information

Finding the basis of the column space is important in understanding the span of vectors and their linear independence. The process helps in various applications, including solving systems of equations and understanding transformations in linear algebra.

Tips

Forgetting to reduce the matrix to row-echelon form, which can lead to incorrectly identifying the pivot columns.
Confusing the pivot columns with non-pivot columns, leading to selecting the wrong basis.
Not selecting the correct columns from the original matrix based on row-reduction results.

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