Transforming an Orthogonal Basis into an Orthonormal Basis
Any orthogonal basis can be transformed into an orthonormal basis through a normalization process.
Difference Between Orthogonal and Orthonormal Bases. An orthonormal basis is an orthogonal basis with orthogonal vectors of unit norm.
How to Normalize an Orthogonal Basis
In the vector space V=R2 over K=R, I have a basis consisting of two vectors.
$$ B = \{ v_1, v_2 \} $$ $$ v_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} $$ $$ v_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix} $$
This is an orthogonal basis because the scalar product of the base vectors taken in pairs equals zero.
$$ \langle v_1, v_2 \rangle = 1 \cdot 1 + 1 \cdot (-1) = 1 - 1 = 0 $$
Is It Also an Orthonormal Basis?
To check if it's also an orthonormal basis, I need to verify if the scalar product of the vectors with themselves equals 1.
$$ \langle v_1, v_1 \rangle = 1 \cdot 1 + 1 \cdot 1 = 1 + 1 = 2 \ne 1 $$ $$ \langle v_2, v_2 \rangle = 1 \cdot 1 + (-1) \cdot (-1) = 1 + 1 = 2 \ne 1 $$
Therefore, the basis B is not an orthonormal basis.
Note. Another way to verify this is through the norm of the vectors. The two vectors are not unit norm vectors, as their Euclidean norm is different from 1 and 0.
$$ ||v_1|| = \sqrt{1^2+1^2} = \sqrt{2} \ne 1 $$ $$ ||v_2|| = \sqrt{1^2+(-1)^2} = \sqrt{2} \ne 1 $$
How to Transform an Orthogonal Basis into an Orthonormal One?
To convert the orthogonal basis into an orthonormal basis, I need to normalize its vectors.
I calculate the Euclidean norm of the vectors that make up the basis B.
$$ ||v_1|| = \sqrt{1^2+1^2} = \sqrt{2} $$ $$ ||v_2|| = \sqrt{1^2+(-1)^2} = \sqrt{2} $$
Then, I divide each vector by its norm, thus obtaining the normalized vectors.
$$ v'_1 = \frac{v_1}{||v_1||}= \frac{ \begin{pmatrix} 1 \\ 1 \end{pmatrix} }{ \sqrt{2} }= \begin{pmatrix} \frac{1}{ \sqrt{2} } \\ \frac{1}{ \sqrt{2} } \end{pmatrix} $$
$$ v'_2 = \frac{v_2}{||v_2||}= \frac{ \begin{pmatrix} 1 \\ -1 \end{pmatrix} }{ \sqrt{2} }= \begin{pmatrix} \frac{1}{ \sqrt{2} \\ \frac{-1}{ \sqrt{2} } } \end{pmatrix} $$
These vectors are the elements of the orthonormal basis B'
$$ B' = \{ v'_1 , v'_2 \} $$
Verification
To check if B' is indeed an orthonormal basis, I calculate the scalar product and the norm of the two normalized vectors.
$$ \langle v'_1,v'_2 \rangle = 0 $$
Since the scalar product equals zero, I have demonstrated that B' is an orthogonal basis.
Now, I calculate the norms of the vectors.
$$ ||v'_1|| = 1 $$ $$ ||v'_2|| = 1 $$
Now, the norms of the vectors are equal to 1.
Therefore, the scalar product of the vectors with themselves is also equal to 1.
$$ \langle v'_1,v'_1 \rangle = 1 $$ $$ \langle v'_2,v'_2 \rangle = 1 $$
This proves that the orthogonal basis B' is also an orthonormal basis.
And so on.
