**Basis of The Dual Space in Finite Dimensions**

Let the vector space *V* have a basis, not necessarily orthogonal. Then the dual space *V** has a basis called the dual basis defined by the special property that

Or, more succinctly,

where δ is the Kronecker delta. Here the superscripts of the basis functionals are not exponents but are instead contravariant indices.

A linear functional belonging to the dual space can be expressed as a linear combination of basis functionals, with coefficients ("components") *u _{i}*,

Then, applying the functional to a basis vector *e _{j}* yields

due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then

that is

This last equation shows that an individual component of a linear functional can be extracted by applying the functional to a corresponding basis vector.

Read more about this topic: Linear Functional, Bases in Finite Dimensions

### Famous quotes containing the words basis of, dimensions, finite, basis, dual and/or space:

“Our fathers and grandfathers who poured over the Midwest were self-reliant, rugged, God-fearing people of indomitable courage.... They asked only for freedom of opportunity and equal chance. In these conceptions lies the real *basis of* American democracy. They and their fathers give a genius to American institutions that distinguished our people from any other in the world.”

—Herbert Hoover (1874–1964)

“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their *dimensions* from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”

—Thomas Jefferson (1743–1826)

“We know then the existence and nature of the *finite*, because we also are *finite* and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”

—Blaise Pascal (1623–1662)

“Buddhists and Christians contrive to agree about death

Making death their ideal *basis* for different ideals.

The Communists however disapprove of death

Except when practical.”

—William Empson (1906–1984)

“Thee for my recitative,

Thee in the driving storm even as now, the snow, the winter-day

declining,

Thee in thy panoply, thy measur’d *dual* throbbing and thy beat

convulsive,

Thy black cylindric body, golden brass and silvery steel,”

—Walt Whitman (1819–1892)

“Play is a major avenue for learning to manage anxiety. It gives the child a safe *space* where she can experiment at will, suspending the rules and constraints of physical and social reality. In play, the child becomes master rather than subject.... Play allows the child to transcend passivity and to become the active doer of what happens around her.”

—Alicia F. Lieberman (20th century)