Linear Space

Question 1

What are axioms of linear space?

Answer 1

• Ax1: x#y=y#x (commutativity)
• Ax2:x#(y#z)=(x#y)#z (associativity)
• Ax3: exist 0(v) € L, such that x#0(v)=x
• Ax4: for every x exist y, such that x#y=0
• Ax5: 1*x=x Ax6 (ab)*x=a*(bx)(the interplay of multiplication)
• Ax7: distributivity
• Ax8: distributivity of * and #

Question 2

What is definition of linear combination?

Answer 2

Let x₁, x₂,…,x_n € L and let a₁,a₂…a_n be scalars in R. Then y defined by the equality y=a₁*x₁#a_{<span>2</span>}*x₂#…#a_n*x_n is called the linear combination of x vectors with a coefficients.

Question 3

What is linear subspace?

Answer 3

Let L be LS and M be a nonvoid subset of L. Then M is called a linear subspace of L if the following 2 cond are satisfied:

• 1) for any pair x€M y€M we have x#y € M
• 2) for any a € R and x € M we have a*x € M

Question 4

Definition of linearly dependent vectors

Answer 4

We have vectors x₁ … x_n in L. If n>=2 then vectors are said to be LD if at least one of them can be written as LC of others.

Question 5

Proposition of linear independence

Answer 5

We have vectors x₁ … x_n. They said to be LI if their only LC = 0 is a trivial combination.

Question 6

Definition of Span.

Answer 6

L is a LS. Let M be a subset of L(m not empty). Then the set of all LC of vectors from M is called a linear span of M(SpanM)

Question 7

What is the definition of linear space?

Answer 7

L is nonvoid set. Let L be equipped with operations #:LxL→L and *:RxL→L such that 8 axioms are satisfied.

Question 8

Defenition of independent subset.

Answer 8

A (possibly infinite) subset M of L is called LI if every choise of finitely many vectors of M gives a LI family in L. If M is not LI, then it’s LD.

Question 9

Defenition of the basis.

Answer 9

Let L be a linear space. A set B, B subset of L, is called a basis of L if B os linearly independent and SpanB=L.

Question 10

Defenition of dimension of LS

Answer 10

Suppose that B={b₁,b₂…b_n} is a basis of L. The number n is called the dimension of L. (dimL=n)

Question 11

Defenition of coordinates in basis.

Answer 11

Let B={b₁,b₂…b_n} be a basis of L. Take a vector x€L. The real numbers x₁,x₃…x_n such that x=x₁*b₁+x₂*b₂+…+x_n*b_n are called the coordinates of x with respect to B. (x=(x₁,x₂,…,x_n)_B)