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Basics of Systems of Linear Equations

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

Basics of Systems of Linear Equations

4 Activities

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

Geometric Interpretation of Linear Systems

3 Activities

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}

More Basics of Systems of Linear Equations Questions:
Which of the following is/are systems of linear equations?
a. 2𝑥1−𝑥3=𝑥22𝑥_1−𝑥_3=𝑥_2
2−𝑥3=02−𝑥_3=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 mm linear equations in nn 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,\ c?
Which of the following is an equation linear in x1, x2, x3x_1,\ x_2,\ x_3?
linear equations
Which of the following equations is/are linear in the variables x, y, zx,\ y,\ z?
i.) 10x+20y2+30z3=010x+20y^2+30z^3=0
ii.) 2x−y3=−1−12z\sqrt{2}x-\frac{y}{3}=-1-\frac{1}{2}z
Which of the following equations is/are linear in the unknowns a,b,ca, b, c ?
i.) ax+by+cz=2ax+by+cz=\sqrt{2}
ii.) ax+2b2y+3c3z=10ax+2b^2y+3c^3z=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=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 = 3
Practice: Linear Equations
For what value(s) of kk will the following equation be linear in x, y, zx,\ y,\ z and a, b, ca,\ b,\ c?
2x+kay−(ln⁡2)b=3−c2x+\frac{k}{ay}-\left(\ln2\right)b=3-c
Which of the following equations is/are linear in the variables x, yx,\ yand zz?
x−y+2z=0x-y+2z=0
x2+2y2+3z2=1x^2+2y^2+3z^2=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}) with mmlinear equations in nnunknowns 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,\ r? [Select all that apply]
Linear algebra
For which values of aaand bbdoes the system 2x−4y=a3x−ay=3b\begin{array}{l}2x-4y = a\\3x-ay = 3b\end{array}have no solutions?
Linear Equations
Which of the following is an equation linear in a, b, ca,\ b,\ c?
Linear Equations
Which of the following is an equation linear in x1,x2,x3x_1, x_2, x_3?
More Geometric Interpretation of Linear Systems Questions:
What is the reduced row echelon form of an augmented matrix with three equations and three unknowns,
a) If all solutions lie on the line[x y z]=(3,1,0)+s(1,2,3)[x\ y\ z] =(3,1,0)+s(1,2,3)
b) If all solutions lie on the plane x+y+2z=1x+y+2z=1
For what value of p does the following system of equations have no solution? What is the geometric interpretation of these equations?
x+3y=1x+3y=1
px−6y=−4px-6y=-4
Practice Question 6: Solving Systems of Linear Equations
Practice Question: Solving Systems of Linear Equations
The following equations represent three planes in ℝ3. Describe the regions of intersection.
a) x1−x2+2x3=0x_1-x_2+2x_3=0
Solving Linear Systems
Practice: Geometric Interpretation of a Linear System
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r} 1&-1&0&0&-5&4\\[0.5em] 0&0&1&0&-2&2\\[0.5em] 0&0&0&1&4&-3 \end{array}\right]
Geometric Interpretation of Linear Systems
Example: Geometric Interpretation of a Linear System
Solve the system of linear equations and interpret the solution geometrically:
2x+2y=6−x−y=−3\begin{array}{rl} 2x+2y&=6\\ -x-y&=-3\\ \end{array}
Homogeneous equations
Find the solution(s) of Ax=0, if the solution of Ax=b is given as follows:
xp=[1030]+s[−1100]+t[00−23]x_p=\begin{bmatrix} 1\\ 0\\ 3\\ 0 \end{bmatrix}+s\begin{bmatrix} -1\\ 1\\ 0\\ 0 \end{bmatrix}+t\begin{bmatrix} 0\\ 0\\ -2\\ 3 \end{bmatrix}
Interpretation of Linear Systems
Which of the following describes the intersection between the following 3 hyperplanes in R4\reals^4?
2x1+3x3+x4=1x2−x4=0x2=−3\begin{array}{rcl} 2x_1+3x_3+x_4&=&1\\ x_2-x_4&=&0\\ x_2&=&-3 \end{array}
The following linear equations represent planes in R3R^3. Describe the intersection of these planes (if any).
2x+y−z=3x−y−z=33x+z=6\begin{array}{c} 2x+y-z=3\\ x-y-z=3\\ 3x+z=6 \end{array}
Homogeneous equations
Find the solution(s) of Ax=0, if the solution of Ax=b is given as follows:
xp=[1030]+s[−1100]+t[00−23]x_p=\begin{bmatrix} 1\\ 0\\ 3\\ 0 \end{bmatrix}+s\begin{bmatrix} -1\\ 1\\ 0\\ 0 \end{bmatrix}+t\begin{bmatrix} 0\\ 0\\ -2\\ 3 \end{bmatrix}
Practice: Solutions to a System of Equations (Step 3)
Practice: Solutions to a System of Equations (Step 3)
Find the solution(s) to the systems of linear equations whose augmented matrices are given:
a.) [1−1030012]\left[ \begin{array}{ccc|c} 1&-1&0&3\\ 0&0&1&2\\ \end{array}\right]
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.
Linear combinations
If u⃗=[5−23]\vec{u}=\begin{bmatrix}5\\-2\\3\end{bmatrix}is a linear combination of v⃗=[−2k3]\vec{v}=\begin{bmatrix}-2\\k\\3\end{bmatrix}and w⃗=[32−1]\vec{w}=\begin{bmatrix}3\\2\\-1\end{bmatrix}, what is the value of kk?
Solving Linear Systems
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r} 1&-1&0&0&-5&4\\[0.5em] 0&0&1&0&-2&2\\[0.5em] 0&0&0&1&4&-3 \end{array}\right]
If the unknowns in this system are x1, x2, x3, x4, x5x_1,\ x_2,\ x_3,\ x_4,\ x_5, find the solution(s) to the linear system.
Solving Linear Systems
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r} 1&-1&0&0&-5&4\\[0.5em] 0&0&1&0&-2&2\\[0.5em] 0&0&0&1&4&-3 \end{array}\right]
If the unknowns in this system are x1, x2, x3, x4, x5x_1,\ x_2,\ x_3,\ x_4,\ x_5, find the solution(s) to the linear system.
Interpretation of Linear Systems
Which of the following describes the intersection between the following 3 hyperplanes in R4\reals^4?
2x1+3x3+x4=1x2−x4=0x2=−3\begin{array}{rcl} 2x_1+3x_3+x_4&=&1\\ x_2-x_4&=&0\\ x_2&=&-3 \end{array}