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Dimension

Let V be a vector space with basis B . The dimension of V , denoted dim V , is the cardinality of B .

Every vector space has a basis.

Every basis for a vector space has the same cardinality.

dim V is well-defined .

If dim V , we say V is finite dimensional .

Examples

vector space field of scalars dimension
N
N
N

Every subspace is a vector space, and therefore has its own dimension.

Suppose S u 1 u k V is a linearly independent set. Then dim < S >

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    Facts

  • If S is a subspace of V , then dim S dim V .
  • If dim S dim V , then S V .

Direct sums

Let V be a vector space, and let S V and T V be subspaces.

We say V is the direct sum of S and T , written V S T , if and only if for every v V , there exist unique s S and t T such that v s t .

If V S T , then T is called a complement of S .

V C { f : | f is continuous } S even funcitons in C T odd funcitons in C f t 1 2 f t f t 1 2 f t f t If f g h g h , g S and g S , h T and h T , then g g h h is odd and even, which implies g g and h h .

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Facts

  • Every subspace has a complement
  • V S T if and only if
    • S T 0
    • < S , T > V
  • If V S T , and dim V , then dim V dim S dim T

Proofs

Invoke a basis.

Norms

Let V be a vector space over F . A norm is a mapping V F , denoted by , such that forall u V , v V , and F

  • u 0 if u 0
  • u u
  • u v u v

Examples

Euclidean norms:

x N : x i 1 N x i 2 1 2 x N : x i 1 N x i 2 1 2

Induced metric

Every norm induces a metric on V d u v u v which leads to a notion of "distance" between vectors.

Inner products

Let V be a vector space over F , F or . An inner product is a mapping V V F , denoted , such that

  • v v 0 , and v v 0 v 0
  • u v v u
  • a u b v w a u w b v w

Examples

N over: x y x y i 1 N x i y i

N over: x y x y i 1 N x i y i

If x x 1 x N , then x x 1 x N is called the "Hermitian," or "conjugatetranspose" of x .

Triangle inequality

If we define u u u , then u v u v Hence, every inner product induces a norm.

Cauchy-schwarz inequality

For all u V , v V , u v u v In inner product spaces, we have a notion of the angle between two vectors: u v u v u v 0 2

Orthogonality

u and v are orthogonal if u v 0 Notation: u v .

If in addition u v 1 , we say u and v are orthonormal .

In an orthogonal (orthonormal) set , each pair of vectors is orthogonal (orthonormal).

Orthogonal vectors in 2 .

Orthonormal bases

An Orthonormal basis is a basis v i such that v i v i i j 1 i j 0 i j

The standard basis for N or N

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The normalized DFT basis u k 1 N 1 2 k N 2 k N N 1

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Expansion coefficients

If the representation of v with respect to v i is v i a i v i then a i v i v

Gram-schmidt

Every inner product space has an orthonormal basis. Any (countable) basis can be made orthogonal by theGram-Schmidt orthogonalization process.

Orthogonal compliments

Let S V be a subspace. The orthogonal compliment S is S u u V u v 0 v v S S is easily seen to be a subspace.

If dim v , then V S S .

If dim v , then in order to have V S S we require V to be a Hilbert Space .

Linear transformations

Loosely speaking, a linear transformation is a mapping from one vector space to another that preserves vector space operations.

More precisely, let V , W be vector spaces over the same field F . A linear transformation is a mapping T : V W such that T a u b v a T u b T v for all a F , b F and u V , v V .

In this class we will be concerned with linear transformations between (real or complex) Euclidean spaces , or subspaces thereof.

Image

T w w W T v w for some v

Nullspace

Also known as the kernel: ker T v v V T v 0

Both the image and the nullspace are easily seen to be subspaces.

Rank

rank T dim T

Nullity

null T dim ker T

Rank plus nullity theorem

rank T null T dim V

Matrices

Every linear transformation T has a matrix representation . If T : 𝔼 N 𝔼 M , 𝔼 or , then T is represented by an M N matrix A a 1 1 a 1 N a M 1 a M N where a 1 i a M i T e i and e i 0 1 0 is the i th standard basis vector.

A linear transformation can be represented with respect to any bases of 𝔼 N and 𝔼 M , leading to a different A . We will always represent a linear transformation using the standard bases.

Column span

colspan A < A > A

Duality

If A : N M , then ker A A

If A : N M , then ker A A

Inverses

The linear transformation/matrix A is invertible if and only if there exists a matrix B such that A B B A I (identity).

Only square matrices can be invertible.

Let A : 𝔽 N 𝔽 N be linear, 𝔽 or . The following are equivalent:

  • A is invertible (nonsingular)
  • rank A N
  • null A 0
  • A 0
  • The columns of A form a basis.

If A A (or A in the complex case), we say A is orthogonal (or unitary ).

Questions & Answers

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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Harper
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s.
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s. Reply
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Graphene has a hexagonal structure
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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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