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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 >

    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 .

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

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

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

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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