Linear Algebra Examples

Step 1
The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre-image of the transformation).
Step 2
Create a system of equations from the vector equation.
Step 3
Write the system of equations in matrix form.
Step 4
Find the reduced row echelon form of the matrix.
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Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Exchange row and row to organize the zeros into position.
Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Tap for more steps...
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Step 5
Use the result matrix to declare the final solutions to the system of equations.
Step 6
This expression is the solution set for the system of equations.
Step 7
Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.
Step 8
Express the vector as a linear combination of column vector using the properties of vector column addition.
Step 9
The null space of the set is the set of vectors created from the free variables of the system.
Step 10
The kernel of is the subspace .
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