System of linear Equations

Question 1

vectors in m-space

(addition, subtraction, multiplication)

Answer 1

the number of components (v₁, v₂, … , v_m)

addition: v+u (v₁ + u₁, v₂ + u₂, … , v_m + u_m)
subtraction: v-u (v₁ - u₁, v₂ - u₂, … , v_m - u_m)
multiplication: cv (cv₁, cv₂, … , cv_m)

Question 2

dot product in m-space

Answer 2

u . v = u₁v₁ + u₂v₂ + … + u_mv_m

Question 3

length of a vector and unit vector in m-space

Answer 3

Question 4

distance of (between) vectors in m-space

Answer 4

Question 5

what is a linear equation?

Answer 5

a linear equation in variables x₁, x₂, … , x_n is an equation that can put in the form

a₁x₁ + a₂x₂ + … + a_nx_n = b

where a₁, a₂, … , a_n and b are constants

ex) linear equations have no squares or varibles dont multiply

x + 2x₂ + … + 10x₁₀ = 20 is a linear equation

x² + y = 3 , x₁x₂x₃ … x₁₀ and xy + z = 1 are not linear equations

Question 6

what is a system of linear equations?

Answer 6

a set of finite linear equations is called a system of linear equations

ex) x₁ + 3x₂ = x₃ and x₂ - x₃ = x₁ or

x + 2y = 1 and x - y = 2 or

x + y = 1, y + z = 2 and x - 2x = 3

the standard form:

1) each equation is writing in the standard form (a₁x₁ + a₂x₂ + … + a_nx_n = b)
2) like variables are aligned a₁₁x₁ + a₁₂x₂ + … + a_1nx_n = b₁

a₂₁x₁ + a₂₂x₂ + … + a_2nx_n = b₂

a_m1x₁ + a_m2x₂ + … + a_mnx_n = b_m

Question 7

augmented matrix of SLE (system of linear equations)

Answer 7

[a₁₁x₁ + a₁₂x₂ + … + a_1nx_n = b₁]

a_m1x₁ + a_m2x₂ + … + a_mnx_n = b_m

from: x₁ + 2x₂ = 3 [1 2 | 3]

x₁ - x₂ = 1 1 -1 | 1

Question 8

what is a solution for SLE (system of linear equations)

Answer 8

A point (s₁, s₂, … , s_n) is a solution for SLE with n-unknown if the point satisfies all the equations of the system

Question 9

geometric meaning of a solution?

Answer 9

every linear equation defines a hyperplane in Rⁿ and a SLE with m equations consists of the hyperplane

any point in the intersection of all these hyperplanes is a solution

infinitely solutions

• > 1-parameter form a line
• > 2-parameters form a plane
• > continues…