Linear Algebra Examples

Find the Eigenvectors/Eigenspace [[3,4],[-2,-3]]
Step 1
Find the eigenvalues.
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Step 1.1
Set up the formula to find the characteristic equation .
Step 1.2
The identity matrix or unit matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere.
Step 1.3
Substitute the known values into .
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Step 1.3.1
Substitute for .
Step 1.3.2
Substitute for .
Step 1.4
Simplify.
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Step 1.4.1
Simplify each term.
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Step 1.4.1.1
Multiply by each element of the matrix.
Step 1.4.1.2
Simplify each element in the matrix.
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Step 1.4.1.2.1
Multiply by .
Step 1.4.1.2.2
Multiply .
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Step 1.4.1.2.2.1
Multiply by .
Step 1.4.1.2.2.2
Multiply by .
Step 1.4.1.2.3
Multiply .
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Step 1.4.1.2.3.1
Multiply by .
Step 1.4.1.2.3.2
Multiply by .
Step 1.4.1.2.4
Multiply by .
Step 1.4.2
Add the corresponding elements.
Step 1.4.3
Simplify each element.
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Step 1.4.3.1
Add and .
Step 1.4.3.2
Add and .
Step 1.5
Find the determinant.
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Step 1.5.1
The determinant of a matrix can be found using the formula .
Step 1.5.2
Simplify the determinant.
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Step 1.5.2.1
Simplify each term.
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Step 1.5.2.1.1
Expand using the FOIL Method.
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Step 1.5.2.1.1.1
Apply the distributive property.
Step 1.5.2.1.1.2
Apply the distributive property.
Step 1.5.2.1.1.3
Apply the distributive property.
Step 1.5.2.1.2
Simplify and combine like terms.
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Step 1.5.2.1.2.1
Simplify each term.
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Step 1.5.2.1.2.1.1
Multiply by .
Step 1.5.2.1.2.1.2
Multiply by .
Step 1.5.2.1.2.1.3
Multiply by .
Step 1.5.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 1.5.2.1.2.1.5
Multiply by by adding the exponents.
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Step 1.5.2.1.2.1.5.1
Move .
Step 1.5.2.1.2.1.5.2
Multiply by .
Step 1.5.2.1.2.1.6
Multiply by .
Step 1.5.2.1.2.1.7
Multiply by .
Step 1.5.2.1.2.2
Add and .
Step 1.5.2.1.2.3
Add and .
Step 1.5.2.1.3
Multiply .
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Step 1.5.2.1.3.1
Multiply by .
Step 1.5.2.1.3.2
Multiply by .
Step 1.5.2.2
Add and .
Step 1.6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 1.7
Solve for .
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Step 1.7.1
Add to both sides of the equation.
Step 1.7.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.7.3
Any root of is .
Step 1.7.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.7.4.1
First, use the positive value of the to find the first solution.
Step 1.7.4.2
Next, use the negative value of the to find the second solution.
Step 1.7.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 2
The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where is the null space and is the identity matrix.
Step 3
Find the eigenvector using the eigenvalue .
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Step 3.1
Substitute the known values into the formula.
Step 3.2
Simplify.
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Step 3.2.1
Subtract the corresponding elements.
Step 3.2.2
Simplify each element.
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Step 3.2.2.1
Subtract from .
Step 3.2.2.2
Subtract from .
Step 3.2.2.3
Subtract from .
Step 3.2.2.4
Subtract from .
Step 3.3
Find the null space when .
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Step 3.3.1
Write as an augmented matrix for .
Step 3.3.2
Find the reduced row echelon form.
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Step 3.3.2.1
Multiply each element of by to make the entry at a .
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Step 3.3.2.1.1
Multiply each element of by to make the entry at a .
Step 3.3.2.1.2
Simplify .
Step 3.3.2.2
Perform the row operation to make the entry at a .
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Step 3.3.2.2.1
Perform the row operation to make the entry at a .
Step 3.3.2.2.2
Simplify .
Step 3.3.3
Use the result matrix to declare the final solution to the system of equations.
Step 3.3.4
Write a solution vector by solving in terms of the free variables in each row.
Step 3.3.5
Write the solution as a linear combination of vectors.
Step 3.3.6
Write as a solution set.
Step 3.3.7
The solution is the set of vectors created from the free variables of the system.
Step 4
Find the eigenvector using the eigenvalue .
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Step 4.1
Substitute the known values into the formula.
Step 4.2
Simplify.
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Step 4.2.1
Add the corresponding elements.
Step 4.2.2
Simplify each element.
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Step 4.2.2.1
Add and .
Step 4.2.2.2
Add and .
Step 4.2.2.3
Add and .
Step 4.2.2.4
Add and .
Step 4.3
Find the null space when .
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Step 4.3.1
Write as an augmented matrix for .
Step 4.3.2
Find the reduced row echelon form.
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Step 4.3.2.1
Multiply each element of by to make the entry at a .
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Step 4.3.2.1.1
Multiply each element of by to make the entry at a .
Step 4.3.2.1.2
Simplify .
Step 4.3.2.2
Perform the row operation to make the entry at a .
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Step 4.3.2.2.1
Perform the row operation to make the entry at a .
Step 4.3.2.2.2
Simplify .
Step 4.3.3
Use the result matrix to declare the final solution to the system of equations.
Step 4.3.4
Write a solution vector by solving in terms of the free variables in each row.
Step 4.3.5
Write the solution as a linear combination of vectors.
Step 4.3.6
Write as a solution set.
Step 4.3.7
The solution is the set of vectors created from the free variables of the system.
Step 5
The eigenspace of is the list of the vector space for each eigenvalue.