Field
The next simplest example is the field F itself. Vector addition is just field addition and scalar multiplication is just field multiplication. Any non-zero element of F serves as a basis so F is a 1-dimensional vector space over itself.
The field is a rather special vector space; in fact it is the simplest example of a commutative algebra over F. Also, F has just two subspaces: {0} and F itself.
Read more about this topic: Examples Of Vector Spaces
Famous quotes containing the word field:
“A field of water betrays the spirit that is in the air. It is continually receiving new life and motion from above. It is intermediate in its nature between land and sky.”
—Henry David Thoreau (1817–1862)
“The snow had begun in the gloaming,
And busily all the night
Had been heaping field and highway
With a silence deep and white.”
—James Russell Lowell (1819–1891)
“Better risk loss of truth than chance of error—that is your faith-vetoer’s exact position. He is actively playing his stake as much as the believer is; he is backing the field against the religious hypothesis, just as the believer is backing the religious hypothesis against the field.”
—William James (1842–1910)