# Linear Algebra Examples

Find an Orthonormal Basis by Gram-Schmidt Method
, ,
Step 1
Assign a name for each vector.
Step 2
The first orthogonal vector is the first vector in the given set of vectors.
Step 3
Use the formula to find the other orthogonal vectors.
Step 4
Find the orthogonal vector .
Step 4.1
Use the formula to find .
Step 4.2
Substitute for .
Step 4.3
Find .
Step 4.3.1
Find the dot product.
Step 4.3.1.1
The dot product of two vectors is the sum of the products of the their components.
Step 4.3.1.2
Simplify.
Step 4.3.1.2.1
Simplify each term.
Step 4.3.1.2.1.1
Multiply by .
Step 4.3.1.2.1.2
Multiply by .
Step 4.3.1.2.1.3
Multiply by .
Step 4.3.1.2.2
Step 4.3.1.2.3
Step 4.3.2
Find the norm of .
Step 4.3.2.1
The norm is the square root of the sum of squares of each element in the vector.
Step 4.3.2.2
Simplify.
Step 4.3.2.2.1
One to any power is one.
Step 4.3.2.2.2
One to any power is one.
Step 4.3.2.2.3
One to any power is one.
Step 4.3.2.2.4
Step 4.3.2.2.5
Step 4.3.3
Find the projection of onto using the projection formula.
Step 4.3.4
Substitute for .
Step 4.3.5
Substitute for .
Step 4.3.6
Substitute for .
Step 4.3.7
Simplify the right side.
Step 4.3.7.1
Rewrite as .
Step 4.3.7.1.1
Use to rewrite as .
Step 4.3.7.1.2
Apply the power rule and multiply exponents, .
Step 4.3.7.1.3
Combine and .
Step 4.3.7.1.4
Cancel the common factor of .
Step 4.3.7.1.4.1
Cancel the common factor.
Step 4.3.7.1.4.2
Rewrite the expression.
Step 4.3.7.1.5
Evaluate the exponent.
Step 4.3.7.2
Multiply by each element of the matrix.
Step 4.3.7.3
Simplify each element in the matrix.
Step 4.3.7.3.1
Multiply by .
Step 4.3.7.3.2
Multiply by .
Step 4.3.7.3.3
Multiply by .
Step 4.4
Substitute the projection.
Step 4.5
Simplify.
Step 4.5.1
Combine each component of the vectors.
Step 4.5.2
Subtract from .
Step 4.5.3
Write as a fraction with a common denominator.
Step 4.5.4
Combine the numerators over the common denominator.
Step 4.5.5
Subtract from .
Step 4.5.6
Write as a fraction with a common denominator.
Step 4.5.7
Combine the numerators over the common denominator.
Step 4.5.8
Subtract from .
Step 5
Find the orthogonal vector .
Step 5.1
Use the formula to find .
Step 5.2
Substitute for .
Step 5.3
Find .
Step 5.3.1
Find the dot product.
Step 5.3.1.1
The dot product of two vectors is the sum of the products of the their components.
Step 5.3.1.2
Simplify.
Step 5.3.1.2.1
Simplify each term.
Step 5.3.1.2.1.1
Multiply by .
Step 5.3.1.2.1.2
Multiply by .
Step 5.3.1.2.1.3
Multiply by .
Step 5.3.1.2.2
Step 5.3.1.2.3
Step 5.3.2
Find the norm of .
Step 5.3.2.1
The norm is the square root of the sum of squares of each element in the vector.
Step 5.3.2.2
Simplify.
Step 5.3.2.2.1
One to any power is one.
Step 5.3.2.2.2
One to any power is one.
Step 5.3.2.2.3
One to any power is one.
Step 5.3.2.2.4
Step 5.3.2.2.5
Step 5.3.3
Find the projection of onto using the projection formula.
Step 5.3.4
Substitute for .
Step 5.3.5
Substitute for .
Step 5.3.6
Substitute for .
Step 5.3.7
Simplify the right side.
Step 5.3.7.1
Rewrite as .
Step 5.3.7.1.1
Use to rewrite as .
Step 5.3.7.1.2
Apply the power rule and multiply exponents, .
Step 5.3.7.1.3
Combine and .
Step 5.3.7.1.4
Cancel the common factor of .
Step 5.3.7.1.4.1
Cancel the common factor.
Step 5.3.7.1.4.2
Rewrite the expression.
Step 5.3.7.1.5
Evaluate the exponent.
Step 5.3.7.2
Multiply by each element of the matrix.
Step 5.3.7.3
Simplify each element in the matrix.
Step 5.3.7.3.1
Multiply by .
Step 5.3.7.3.2
Multiply by .
Step 5.3.7.3.3
Multiply by .
Step 5.4
Find .
Step 5.4.1
Find the dot product.
Step 5.4.1.1
The dot product of two vectors is the sum of the products of the their components.
Step 5.4.1.2
Simplify.
Step 5.4.1.2.1
Simplify each term.
Step 5.4.1.2.1.1
Multiply .
Step 5.4.1.2.1.1.1
Multiply by .
Step 5.4.1.2.1.1.2
Multiply by .
Step 5.4.1.2.1.2
Multiply by .
Step 5.4.1.2.1.3
Multiply by .
Step 5.4.1.2.2
Step 5.4.1.2.3
Step 5.4.2
Find the norm of .
Step 5.4.2.1
The norm is the square root of the sum of squares of each element in the vector.
Step 5.4.2.2
Simplify.
Step 5.4.2.2.1
Use the power rule to distribute the exponent.
Step 5.4.2.2.1.1
Apply the product rule to .
Step 5.4.2.2.1.2
Apply the product rule to .
Step 5.4.2.2.2
Raise to the power of .
Step 5.4.2.2.3
Multiply by .
Step 5.4.2.2.4
Raise to the power of .
Step 5.4.2.2.5
Raise to the power of .
Step 5.4.2.2.6
Apply the product rule to .
Step 5.4.2.2.7
One to any power is one.
Step 5.4.2.2.8
Raise to the power of .
Step 5.4.2.2.9
Apply the product rule to .
Step 5.4.2.2.10
One to any power is one.
Step 5.4.2.2.11
Raise to the power of .
Step 5.4.2.2.12
Combine the numerators over the common denominator.
Step 5.4.2.2.13
Step 5.4.2.2.14
Combine the numerators over the common denominator.
Step 5.4.2.2.15
Step 5.4.2.2.16
Cancel the common factor of and .
Step 5.4.2.2.16.1
Factor out of .
Step 5.4.2.2.16.2
Cancel the common factors.
Step 5.4.2.2.16.2.1
Factor out of .
Step 5.4.2.2.16.2.2
Cancel the common factor.
Step 5.4.2.2.16.2.3
Rewrite the expression.
Step 5.4.2.2.17
Rewrite as .
Step 5.4.2.2.18
Multiply by .
Step 5.4.2.2.19
Combine and simplify the denominator.
Step 5.4.2.2.19.1
Multiply by .
Step 5.4.2.2.19.2
Raise to the power of .
Step 5.4.2.2.19.3
Raise to the power of .
Step 5.4.2.2.19.4
Use the power rule to combine exponents.
Step 5.4.2.2.19.5
Step 5.4.2.2.19.6
Rewrite as .
Step 5.4.2.2.19.6.1
Use to rewrite as .
Step 5.4.2.2.19.6.2
Apply the power rule and multiply exponents, .
Step 5.4.2.2.19.6.3
Combine and .
Step 5.4.2.2.19.6.4
Cancel the common factor of .
Step 5.4.2.2.19.6.4.1
Cancel the common factor.
Step 5.4.2.2.19.6.4.2
Rewrite the expression.
Step 5.4.2.2.19.6.5
Evaluate the exponent.
Step 5.4.2.2.20
Simplify the numerator.
Step 5.4.2.2.20.1
Combine using the product rule for radicals.
Step 5.4.2.2.20.2
Multiply by .
Step 5.4.3
Find the projection of onto using the projection formula.
Step 5.4.4
Substitute for .
Step 5.4.5
Substitute for .
Step 5.4.6
Substitute for .
Step 5.4.7
Simplify the right side.
Step 5.4.7.1
Simplify the denominator.
Step 5.4.7.1.1
Apply the product rule to .
Step 5.4.7.1.2
Rewrite as .
Step 5.4.7.1.2.1
Use to rewrite as .
Step 5.4.7.1.2.2
Apply the power rule and multiply exponents, .
Step 5.4.7.1.2.3
Combine and .
Step 5.4.7.1.2.4
Cancel the common factor of .
Step 5.4.7.1.2.4.1
Cancel the common factor.
Step 5.4.7.1.2.4.2
Rewrite the expression.
Step 5.4.7.1.2.5
Evaluate the exponent.
Step 5.4.7.1.3
Raise to the power of .
Step 5.4.7.1.4
Cancel the common factor of and .
Step 5.4.7.1.4.1
Factor out of .
Step 5.4.7.1.4.2
Cancel the common factors.
Step 5.4.7.1.4.2.1
Factor out of .
Step 5.4.7.1.4.2.2
Cancel the common factor.
Step 5.4.7.1.4.2.3
Rewrite the expression.
Step 5.4.7.2
Multiply the numerator by the reciprocal of the denominator.
Step 5.4.7.3
Cancel the common factor of .
Step 5.4.7.3.1
Cancel the common factor.
Step 5.4.7.3.2
Rewrite the expression.
Step 5.4.7.4
Multiply by each element of the matrix.
Step 5.4.7.5
Simplify each element in the matrix.
Step 5.4.7.5.1
Cancel the common factor of .
Step 5.4.7.5.1.1
Move the leading negative in into the numerator.
Step 5.4.7.5.1.2
Factor out of .
Step 5.4.7.5.1.3
Cancel the common factor.
Step 5.4.7.5.1.4
Rewrite the expression.
Step 5.4.7.5.2
Move the negative in front of the fraction.
Step 5.4.7.5.3
Multiply .
Step 5.4.7.5.3.1
Multiply by .
Step 5.4.7.5.3.2
Multiply by .
Step 5.4.7.5.4
Multiply .
Step 5.4.7.5.4.1
Multiply by .
Step 5.4.7.5.4.2
Multiply by .
Step 5.5
Substitute the projections.
Step 5.6
Simplify.
Step 5.6.1
Combine each component of the vectors.
Step 5.6.2
Combine each component of the vectors.
Step 5.6.3
Multiply .
Step 5.6.3.1
Multiply by .
Step 5.6.3.2
Multiply by .
Step 5.6.4
Combine fractions.
Step 5.6.4.1
Combine the numerators over the common denominator.
Step 5.6.4.2
Simplify the expression.
Step 5.6.4.2.1
Step 5.6.4.2.2
Divide by .
Step 5.6.5
Multiply by .
Step 5.6.6
Subtract from .
Step 5.6.7
To write as a fraction with a common denominator, multiply by .
Step 5.6.8
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 5.6.8.1
Multiply by .
Step 5.6.8.2
Multiply by .
Step 5.6.9
Simplify the expression.
Step 5.6.9.1
Combine the numerators over the common denominator.
Step 5.6.9.2
Subtract from .
Step 5.6.10
Cancel the common factor of and .
Step 5.6.10.1
Factor out of .
Step 5.6.10.2
Cancel the common factors.
Step 5.6.10.2.1
Factor out of .
Step 5.6.10.2.2
Cancel the common factor.
Step 5.6.10.2.3
Rewrite the expression.
Step 5.6.11
Move the negative in front of the fraction.
Step 5.6.12
Find the common denominator.
Step 5.6.12.1
Write as a fraction with denominator .
Step 5.6.12.2
Multiply by .
Step 5.6.12.3
Multiply by .
Step 5.6.12.4
Multiply by .
Step 5.6.12.5
Multiply by .
Step 5.6.12.6
Reorder the factors of .
Step 5.6.12.7
Multiply by .
Step 5.6.13
Combine the numerators over the common denominator.
Step 5.6.14
Simplify by subtracting numbers.
Step 5.6.14.1
Subtract from .
Step 5.6.14.2
Subtract from .
Step 5.6.15
Cancel the common factor of and .
Step 5.6.15.1
Factor out of .
Step 5.6.15.2
Cancel the common factors.
Step 5.6.15.2.1
Factor out of .
Step 5.6.15.2.2
Cancel the common factor.
Step 5.6.15.2.3
Rewrite the expression.
Step 6
Find the orthonormal basis by dividing each orthogonal vector by its norm.
Step 7
Find the unit vector where .
Step 7.1
To find a unit vector in the same direction as a vector , divide by the norm of .
Step 7.2
The norm is the square root of the sum of squares of each element in the vector.
Step 7.3
Simplify.
Step 7.3.1
One to any power is one.
Step 7.3.2
One to any power is one.
Step 7.3.3
One to any power is one.
Step 7.3.4
Step 7.3.5
Step 7.4
Divide the vector by its norm.
Step 7.5
Divide each element in the vector by .
Step 8
Find the unit vector where .
Step 8.1
To find a unit vector in the same direction as a vector , divide by the norm of .
Step 8.2
The norm is the square root of the sum of squares of each element in the vector.
Step 8.3
Simplify.
Step 8.3.1
Use the power rule to distribute the exponent.
Step 8.3.1.1
Apply the product rule to .
Step 8.3.1.2
Apply the product rule to .
Step 8.3.2
Raise to the power of .
Step 8.3.3
Multiply by .
Step 8.3.4
Raise to the power of .
Step 8.3.5
Raise to the power of .
Step 8.3.6
Apply the product rule to .
Step 8.3.7
One to any power is one.
Step 8.3.8
Raise to the power of .
Step 8.3.9
Apply the product rule to .
Step 8.3.10
One to any power is one.
Step 8.3.11
Raise to the power of .
Step 8.3.12
Combine the numerators over the common denominator.
Step 8.3.13
Step 8.3.14
Combine the numerators over the common denominator.
Step 8.3.15
Step 8.3.16
Cancel the common factor of and .
Step 8.3.16.1
Factor out of .
Step 8.3.16.2
Cancel the common factors.
Step 8.3.16.2.1
Factor out of .
Step 8.3.16.2.2
Cancel the common factor.
Step 8.3.16.2.3
Rewrite the expression.
Step 8.3.17
Rewrite as .
Step 8.4
Divide the vector by its norm.
Step 8.5
Divide each element in the vector by .
Step 8.6
Simplify.
Step 8.6.1
Multiply the numerator by the reciprocal of the denominator.
Step 8.6.2
Multiply by .
Step 8.6.3
Move to the left of .
Step 8.6.4
Move to the left of .
Step 8.6.5
Multiply the numerator by the reciprocal of the denominator.
Step 8.6.6
Multiply by .
Step 8.6.7
Multiply the numerator by the reciprocal of the denominator.
Step 8.6.8
Multiply by .
Step 9
Find the unit vector where .
Step 9.1
To find a unit vector in the same direction as a vector , divide by the norm of .
Step 9.2
The norm is the square root of the sum of squares of each element in the vector.
Step 9.3
Simplify.
Step 9.3.1
Raising to any positive power yields .
Step 9.3.2
Use the power rule to distribute the exponent.
Step 9.3.2.1
Apply the product rule to .
Step 9.3.2.2
Apply the product rule to .
Step 9.3.3
Raise to the power of .
Step 9.3.4
Multiply by .
Step 9.3.5
One to any power is one.
Step 9.3.6
Raise to the power of .
Step 9.3.7
Apply the product rule to .
Step 9.3.8
One to any power is one.
Step 9.3.9
Raise to the power of .
Step 9.3.10
Step 9.3.11
Combine the numerators over the common denominator.
Step 9.3.12
Step 9.3.13
Cancel the common factor of and .
Step 9.3.13.1
Factor out of .
Step 9.3.13.2
Cancel the common factors.
Step 9.3.13.2.1
Factor out of .
Step 9.3.13.2.2
Cancel the common factor.
Step 9.3.13.2.3
Rewrite the expression.
Step 9.3.14
Rewrite as .
Step 9.3.15
Any root of is .
Step 9.4
Divide the vector by its norm.
Step 9.5
Divide each element in the vector by .
Step 9.6
Simplify.
Step 9.6.1
Multiply the numerator by the reciprocal of the denominator.
Step 9.6.2
Multiply by .
Step 9.6.3
Multiply the numerator by the reciprocal of the denominator.
Step 9.6.4
Combine and .
Step 9.6.5
Multiply the numerator by the reciprocal of the denominator.
Step 9.6.6
Combine and .
Step 10
Substitute the known values.