How to find basis for null space?
Understand the Problem
The question is asking for the method to determine the basis for the null space of a given matrix. The null space consists of all solutions to the equation Ax = 0, where A is the matrix. To find the basis, one generally performs row reduction to echelon form and solves for the free variables to express the solutions in terms of these variables.
Answer
The basis for the null space is found by row reducing matrix $A$ and expressing pivot variables in terms of free variables to form solution vectors.
Answer for screen readers
The basis for the null space of matrix $A$ consists of the solution vectors derived from the reduced row echelon form of $A$. These vectors represent all solutions to the equation $Ax = 0$.
Steps to Solve
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Set up the equation We start with the matrix equation $Ax = 0$, where $A$ is your given matrix and $x$ is the vector of variables. For example, if $A$ is a $3 \times 3$ matrix, then $x$ will be a column vector with 3 variables.
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Row reduce the matrix Use row operations to reduce the matrix $A$ to its row echelon form or reduced row echelon form (RREF). This will help us see the relationships between the variables more clearly.
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Identify the pivots and free variables Once in row echelon form, identify the pivot columns (columns with leading 1s) and the free columns (columns without leading 1s). The pivot variables correspond to the pivot columns, while the free variables can take on any value.
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Express pivot variables in terms of free variables Using the row-reduced form of the matrix, express each pivot variable in terms of the free variables. This step enables you to construct the general solution to the equation $Ax = 0$.
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Construct the solution vectors For each free variable, create a solution vector by setting one free variable to 1 and all others to 0. These vectors will form the basis for the null space.
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Form the basis of the null space The set of solution vectors formed in the previous step constitutes the basis for the null space. List these vectors together, and you have the null space basis.
The basis for the null space of matrix $A$ consists of the solution vectors derived from the reduced row echelon form of $A$. These vectors represent all solutions to the equation $Ax = 0$.
More Information
The basis of the null space represents all linear combinations of vectors that, when multiplied by the matrix $A$, yield the zero vector. This is crucial in understanding the linear transformations represented by $A$ and their effects on vector spaces.
Tips
- Not performing the row reduction correctly: Ensure row operations are performed accurately.
- Misidentifying pivot and free variables: Double-check to ensure pivot columns are identified correctly and that free variables are noted.
- Forgetting to express all pivot variables in terms of free variables: Ensure all relationships are captured systematically to avoid leaving out important dependencies.
