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Basics of Systems of Linear Equations

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Basics of Systems of Linear Equations
Linear Equations
A linear equation with unknowns x1,x2,…,xn\colorOne{x_1,x_2,\dots,x_n}x1​,x2​,…,xn​ is an equation written in the form:
𝑎1𝑥1+𝑎2𝑥2+⋯+𝑎𝑛𝑥𝑛=𝑏\boxed{\quad
𝑎_1\colorOne{𝑥_1}+𝑎_2\colorOne{𝑥_2}+⋯+𝑎_𝑛\colorOne{𝑥_𝑛}=𝑏
\quad}a1​x1​+a2​x2​+⋯+an​xn​=b​
Notes
Each term is of degree 1   ⟹  \implies⟹ a coefficient ai∈Ra_i \in \realsai​∈R times a single variable/unknown
aia_iai​ cannot all be 0

Examples
linear equation with 2 unknowns:  ln⁡(3)x−1πy=1\ln(3)x-\dfrac{1}{\pi}y=1ln(3)x−π1​y=1
linear equation with 4 unknowns: −4𝑥1+5𝑥2−x3+2𝑥4=e2-4𝑥_1 + 5𝑥_2 - x_3 + 2𝑥_4 = e^2−4x1​+5x2​−x3​+2x4​=e2
non-linear equation with 2 unknowns:  sin⁡(3x)−y2=1\sin(3x)-y^2=1sin(3x)−y2=1

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System of Linear Equations
A system of linear equations (SLE) or linear system is a collection of linear equations with the same variables x1,x2,…,xn\colorOne{x_1,x_2,\dots,x_n}x1​,x2​,…,xn​:
m equations {a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋮am1x1+am2x2+⋯+amnxn=bmm \text{ equations }
\left\{
\begin{array}{ccccccccl}
  a_{11}\colorOne{\colorOne{x_1}}&+&a_{12}\colorOne{\colorOne{x_2}}&+&\cdots&+&a_{1n}\colorOne{\colorOne{x_n}} &=& b_1\\
  a_{21}\colorOne{x_1}&+&a_{22}\colorOne{x_2}&+&\cdots&+&a_{2n}\colorOne{x_n} &=& b_2\\
  &&&\vdots&&\\
    a_{m1}\colorOne{x_1}&+&a_{m2}\colorOne{x_2}&+&\cdots&+&a_{mn}\colorOne{x_n} &=& b_m\\
\end{array}
\right.m equations ⎩⎨⎧​a11​x1​a21​x1​am1​x1​​+++​a12​x2​a22​x2​am2​x2​​++⋮+​⋯⋯⋯​+++​a1n​xn​a2n​xn​amn​xn​​===​b1​b2​bm​​

A system with mmm equations (rows) and nnn unknowns (terms/columns) is of size m×nm\times nm×n.

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Solutions to Linear Systems
A solution to an m×nm\times nm×n  SLE is a vector x⃗=⟨x1,x2,…,xn⟩\vec x = \lang x_1, x_2, \dots, x_n\rangx=⟨x1​,x2​,…,xn​⟩ that satisfies all mmm equations.
Two SLEs are equivalent if they have the same solution.
Example
x⃗=[xy]=[1−2]\vec x 
= 
\begin{bmatrix}
  x\\ y
\end{bmatrix}
= 
\begin{bmatrix}
  1\\ -2
\end{bmatrix}x=[xy​]=[1−2​] is a solution to the following equivalent SLEs:
  x+y=−12x−y=4\begin{array}{rcl}
x+y&=&-1\\
2x-y&=&4
\end{array}x+y2x−y​==​−14​   and   2x+2y=−22x−y=4\begin{array}{rcl}
2x+2y&=&-2\\
2x-y&=&4
\end{array}2x+2y2x−y​==​−24​
Family of Solutions
A k\colorFour{\bm{k}}k-parameter family of solutions to a SLE is a set of solutions which contains kkk parameters (real number variables).
A family of solutions consists of infinitely many solutions (but not every vector is a solution!)
Example
x⃗=[1t2−ss]\vec x = 
\begin{bmatrix}
  1\\ t\\ 2-s\\ s
\end{bmatrix}x=​1t2−ss​​ is a 2-parameter family of solutions (with parameters s, t∈Rs,\ t \in \realss, t∈R) to the SLE:
x1+x3+x4=3x1−x3−x4=−1\begin{aligned}
x_1 +x_3+x_4&=3\\
x_1 -x_3-x_4 &=-1
\end{aligned}x1​+x3​+x4​x1​−x3​−x4​​=3=−1​
Wize Tip
We can rewrite the general solution to clearly see that it is a 2-parameter family of solutions:
x⃗=[1020]+t[0100]+s[00−11]\vec x = 
\begin{bmatrix}
  1\\ 0\\ 2\\ 0
\end{bmatrix}
+
t\begin{bmatrix}
  0\\ 1\\ 0\\ 0
\end{bmatrix}
+s\begin{bmatrix}
  0\\ 0\\ -1\\ 1
\end{bmatrix}x=​1020​​+t​0100​​+s​00−11​​ 

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Example: Linear Equations
1. Determine whether the following are linear equations in the variables x, y, zx,\ y,\ zx, y, z.
A)  5x−y2=3−0.5z\sqrt{5}x-\frac{y}{2}=3-0.5z5​x−2y​=3−0.5z

Linear. (Can be rearranged to the standard form)
The variables x, y, zx,\ y,\ zx, y, z are all linear with constant coefficients in front.

B)  2𝑥2−sin⁡𝑦+1𝑧=02𝑥^2−\sin𝑦+\frac1𝑧=02x2−siny+z1​=0

Non-linear.
x2x^2x2 is quadratic, sin⁡ y\sin\ ysin y is a trigonometric function, 1z\frac{1}{z}z1​ is a reciprocal function.

C)  sin⁡(40)𝑥+24𝑧=−13\sin(40)𝑥+2^4𝑧=−\frac{1}{3}sin(40)x+24z=−31​

Linear.
x, zx,\ zx, z are linear with constant coefficients in front, and the coefficient in front of yyy is 0.

D)  sin⁡(0)x+(e0−1)z=0\sin(0)x + (e^0 - 1)z = 0sin(0)x+(e0−1)z=0

Non-linear.
The coefficients for each of the variables x, y, zx,\ y,\ zx, y, z is 0 (yyy does not appear,  sin⁡(0)=0\sin(0)=0sin(0)=0, and e0−1=0e^0 -1 = 0e0−1=0).
This equation just says 0=00=00=0, and since there are no variables, it is not linear.

Which of the following are linear equations in the variables p, q, rp,\ q,\ rp, q, r? [Select all that apply]

A) xp+y𝑞2+z𝑟3=3 xp + y𝑞^2 + z𝑟^3 = 3xp+yq2+zr3=3

B) 𝑝𝑥−𝑞𝑦−𝑧=0 \sqrt{𝑝𝑥} − 𝑞𝑦 − 𝑧 = 0px​−qy−z=0

C) 𝑥2p+yq − 𝑟𝑧=10𝑥^{2}p+yq\ −\ \dfrac{𝑟}{𝑧}=10x2p+yq − zr​=10

D) 𝑝𝑥=0𝑝𝑥=0px=0

I don't know

Determine which vectors are solutions to the following SLE [select all that apply]:
x1 − 5x2 +x3=2−x1 +5x2 −3x3= −6\begin{alignedat}{8}
x_1\ &-&\ 5x_2\ &+& x_3 &=& 2\\
-x_1\ &+& 5x_2\ &-& 3x_3 &=&\ -6
\end{alignedat}x1​ −x1​ ​−+​ 5x2​ 5x2​ ​+−​x3​3x3​​==​2 −6​

A)  a⃗=[1t−1]\vec a =
\begin{bmatrix}
  1\\ t\\ -1 
\end{bmatrix}
a=​1t−1​​

B)  b⃗=[5tt2]\vec b =
\begin{bmatrix}
  5t\\ t\\ 2 
\end{bmatrix}
b=​5tt2​​

C)  c⃗=[101]\vec c =
\begin{bmatrix}
  1\\ 0\\ 1 
\end{bmatrix}c=​101​​

D)  d⃗=[−5−12]\vec d =
\begin{bmatrix}
  -5\\ -1\\ 2 
\end{bmatrix}d=​−5−12​​

I don't know

Extra Practice

Which of the following is/are systems of linear equations?
a. 2𝑥1−𝑥3=𝑥22𝑥_1−𝑥_3=𝑥_22x1​−x3​=x2​
       2−𝑥3=02−𝑥_3=02−x3​=0

Which of the following is an equation linear in 𝑎,𝑏,𝑐?

Which of the following is an equation linear in 𝑥1, 𝑥2, 𝑥3?

Consider a system of mmm linear equations in nnn variables. If the solution has at least 3 parameters, then it must be true that

Which of the following is an equation linear in a, b, ca,\ b,\ ca, b, c?

Which of the following is an equation linear in x1, x2, x3x_1,\ x_2,\ x_3x1​, x2​, x3​?

linear equations

Which of the following equations is/are linear in the variables x, y, zx,\ y,\ zx, y, z?
i.) 10x+20y2+30z3=010x+20y^2+30z^3=010x+20y2+30z3=0
ii.) 2x−y3=−1−12z\sqrt{2}x-\frac{y}{3}=-1-\frac{1}{2}z2​x−3y​=−1−21​z

Which of the following equations is/are linear in the unknowns a,b,ca, b, c
a,b,c?
i.) ax+by+cz=2ax+by+cz=\sqrt{2}ax+by+cz=2​
ii.) ax+2b2y+3c3z=10ax+2b^2y+3c^3z=10ax+2b2y+3c3z=10

Practice Question: Linear Equations

Practice Question: Linear Equations
Which of the following is/are linear equations in the variables 𝑥, 𝑦, 𝑧 and in 𝑎, 𝑏, 𝑐?
i. 𝑎𝑥+𝑏𝑦+𝑐𝑧=5𝑎𝑥+𝑏𝑦+𝑐𝑧=5ax+by+cz=5

Practice: Linear Equations (~2016 Test2 #1)

Practice: Linear Equations
Which of the following is/are equations linear in 𝑝, 𝑞, 𝑟? (select all that apply)
a)𝑝𝑥+𝑞2𝑦+𝑟3𝑧=3a) 𝑝𝑥 + 𝑞^2𝑦 + 𝑟^3𝑧 = 3a)px+q2y+r3z=3

Practice: Linear Equations

For what value(s) of kkk will the following equation be linear in x, y, zx,\ y,\ zx, y, z and a, b, ca,\ b,\ ca, b, c?
2x+kay−(ln⁡2)b=3−c2x+\frac{k}{ay}-\left(\ln2\right)b=3-c2x+ayk​−(ln2)b=3−c

 Which of the following equations is/are linear in the variables x, yx,\ yx, yand zzz?
x−y+2z=0x-y+2z=0x−y+2z=0
x2+2y2+3z2=1x^2+2y^2+3z^2=1x2+2y2+3z2=1

System of Equations

e.g. True or False: if a linear system has more unknowns than equations, then it must have infinitely many solutions. Justify your answer with a proof or counter-example.

A homogeneous system (Ax⃗=0⃗A\vec{x}=\vec{0}Ax=0) with mmmlinear equations in nnnunknowns has exactly one solution. Which of the following statements is always true?

System of Equations

e.g. True or False: if a linear system has more unknowns than equations, then it must have infinitely many solutions. Justify your answer with a proof or counter-example.

System of Equations

True or False: if a linear system has more unknowns than equations, then it must have infinitely many solutions. Justify your answer with a proof or counter-example.

Basics of Systems of Linear Equations

Which of the following are linear equations in the variables p, q, rp,\ q,\ rp, q, r? [Select all that apply]

Linear algebra

For which values of aaaand bbbdoes the system 2x−4y=a3x−ay=3b\begin{array}{l}2x-4y = a\\3x-ay = 3b\end{array}2x−4y=a3x−ay=3b​have no solutions?

Linear Equations

Which of the following is an equation linear in a, b, ca,\ b,\ ca, b, c?

Linear Equations

Which of the following is an equation linear in x1,x2,x3x_1, x_2, x_3x1​,x2​,x3​?

Basics of Systems of Linear Equations

Determine which vectors are solutions to the following SLE [select all that apply]:
x1 − 5x2 +x3=2−x1 +5x2 −3x3= −6\begin{alignedat}{8}
x_1\ &-&\ 5x_2\ &+& x_3 &=& 2\\
-x_1\ &+& 5x_2\ &-& 3x_3 &=&\ -6
\end{alignedat}x1​ −x1​ ​−+​ 5x2​ 5x2​ ​+−​x3​3x3​​==​2 −6​

Wize University Linear Algebra Textbook > Systems of Linear Equations (SLEs) (Linear Systems)

Basics of Systems of Linear Equations

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