Orthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
Intuitive overview
The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero.
Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian space, the norm of a vector is the square root of the vector dotted with itself. That is,
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by Goo Goo Dolls
We Are The Normal
by: Goo Goo DollsIt's a beautiful day, I heard everybody say
The sun shines down for all of us
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Even still, I like mine with a chill
That ain't so obvious
We are the normal
We live and we die
With no reason why
We are the normal
We live and we die
With no reason why
It's a beautiful life, and I've got it in my sights
And that ain't for all of us
It's a beautiful sound
When my life comes crashin' down
That ain't so obvious
It's a beautiful sin, and I'm doin' it again
And that ain't for all of us
It's a beautiful life, and I've got it in my sights
Ain't that so obvious?
We are the normal
We live and we die
With no reason why
We are the normal
We live and we die
With no reason why
We are the normal
We live and we die
With no reason why
We are the normal
We live and we die
With no reason why
There's no reason why
We are the normal
We live and we die
With no reason why
We are the normal
We live and we die