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Interpretation of Linear Systems

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

Geometric Interpretation of Linear Systems

3 Activities

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

Solving Linear Systems

6 Activities

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}
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}
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.
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}
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}
More Solving Linear Systems Questions:
Find size of soln set to nonhom linear system
Find all solutions to the system of linear equations 5x+y+z=0x+2z=−12x+y−5z=1\begin{array}{rcl} 5x+y+z=0\\ x+2z=-1\\ 2x+y-5z=1 \end{array}.
Given the augmented matrix [1201010200k2−9k−3]\left[\begin{array}{rrr|r} 1 & 2 &0 &1 \\ 0 & 1 &0&2\\ 0&0&k^2-9&k-3 \end{array}\right], answer the following questions.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 44.4
Interpret the following matrix as the RREF of a matrix of coefficients augmented by the vector of constants and determine whether the systems has a unique solution.
[100001000010] \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}
Find the all values of p and q such that the system has at least one solution.
[12401300p∣15q]\left[\begin{array}{l} 1&2&4\\ 0&1&3\\ 0&0&p\\ \end{array}\right.\left|\begin{array}{l}1 \\5\\q\end{array}\right]
Find the value of p such that there is a unique solution to this system of equations:
L:{         2x1+x3=4x1+3x2+px3=−10     x1−x2+3x3=5L:\begin{cases} \ \ \ \ \ \ \ \ \ 2x_1+x_3=4\\ x_1+3x_2+px_3=-10\\ \ \ \ \ \ x_1-x_2+3x_3=5 \end{cases}
a) For what values of c does the system have one solution?
{2𝑥+𝒄𝑦=0𝒄𝑥+𝑦=3\begin{cases} 2𝑥 + 𝒄𝑦 = 0\\ 𝒄𝑥 + 𝑦 = 3 \end{cases}
b) For what values of c does it have infinitely many solutions?
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: Solving a System of Linear Equations (Infinitely Many Solutions)
Practice Question: Solving a SLE
Solve the following system of linear equations: x+2y=4y+z=3z+w=1\begin{array}{rcl} x+2y&=&4\\ y+z&=&3\\ z+w&=&1 \end{array}.
The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.
Which of the following options is the general solution of the following system of linear equations?
x−y−z=2x+y+z=0\begin{array}{l}x-y-z=2\\x+y+z=0\end{array}
Find the basic solutions of the following homogeneous system of linear equations:
x1−2x2+3x3+x4+2x5=0x1−x2+3x3−x4+3x5=0−x1+2x2−4x3+x4+x5=0\begin{array}{l}x_1 - 2x_2 + 3x_3 + x_4 + 2x_5 = 0\\x_1 - x_2 + 3x_3 - x_4 + 3x_5 = 0\\-x_1 + 2x_2 - 4x_3 + x_4 + x_5 = 0\end{array}
What can you say about the following systems of linear equations?
x−2y−3z=42x+2y−6z=−1\begin{array}{l}x-2y-3z=4\\2x+2y-6z=-1\end{array}
What can you say about the following systems of linear equations?
x−y+3z=52x−2y+6z=−10\begin{array}{l}x-y+3z=5\\2x-2y+6z=-10\end{array}
Write the general solution to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l} x - y - 3z = -4\\ 2x - 3y + z = 0 \end{array}
Wize Concept Clarifier--Gaussian Jordan Elimination
Wize Concept Clarifier
Solve the system x+y−z=24y−4z=12x+3y−3z=−1\begin{array}{} x+y-z=2\\ 4y-4z=1\\ 2x+3y-3z=-1 \end{array}.
The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.
Linear Systems - Mixing
You are the cook on a celebrity yacht, and you have run out of flour, sugar, and baking powder -
except for that contained in a large quantity of 3 different pancake mixes:
Mix A: 8 parts flour to 8 parts sugar to 1 part baking powder
Basics of Systems of Linear Equations
Practice: Solutions to Linear Systems
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}
123 - FML #1 - 18.1W - e.g. 20
Consider the following system of linear equations:
{x−2y+4z=22x+y−2z=−13x−y+2z=12x+6y−12z=−6\bcb{ \begin{cases} x - 2y + 4z = 2& \\ 2x + y - 2z = -1 &\\ 3x - y + 2z = 1 &\\ 2x + 6y - 12z = -6& \end{cases}}
a) Construct the augmented matrix A\bcb{A} corresponding to this system.
Solve linear systems
Solve the following systems of linear equations:
a) x+2y=−3−x+3y=1\begin{array}{r}x+2y=-3\\-x+3y=1\end{array}
b) x+y−2z=1−4x+3y+2z=1−3x+11x−6z=0\begin{array}{r} x+y-2z=1\\ -4x+3y+2z=1\\ -3x+11x-6z=0 \end{array}
123 - FML #1 - 18.1W - e.g. 21.1
Solve the following system of linear equations:
{2x+y+z+3w=13x−2z+2w=−15x+y−z+5w=07x+2y+8w=1\bcb{ \begin{cases} 2x + y + z + 3w = 1 \\ 3x - 2z + 2w = -1 \\ 5x + y - z + 5w = 0 \\ 7x + 2y + 8w = 1 \end{cases} }
Solve the following linear system of equations by Gaussian Elimination.
2x−y+5z=14x+y+z=4−x−3y+3z=−2\begin{array}{c} 2x - y + 5z = 14\\ x + y + z = 4\\ -x - 3y + 3z = - 2 \end{array}
Solve the following linear system of equations.
3x+2y−2z=185x+2y−z=27x+y+z=9\begin{array}{c} 3x + 2y - 2z = 18\\ 5x + 2y - z =27\\ x + y + z = 9 \end{array}
Find soln to nonhom linear system
Find all solutions to the following system of linear equations[10210131]\left[\begin{array}{rrr|r} 1&0&2&1\\ 0&1&3&1 \end{array}\right].
Balancing Chemical Equations
The chemical equation for the combustion of octane (the primary component of gasoline) is:
C8H18+O2=H2O+CO2C_8H_{18}+O_2 = H_2O+CO_2
Balance this equation using Gauss-Jordan elimination
Find all possible solutions to the following linear system of equations.
x+2y−z=9x−y+2z=3\begin{array}{c} x + 2y - z = 9\\ x - y + 2z = 3 \end{array}
Solving Linear Systems
Practice: Solving a Linear System
Solve the following system of linear equations. What type of solution is it?
3x1+6x2=12x1+4x2=2x2+x3=3\begin{aligned} 3x_1 + 6x_2&= 1\\ 2x_1 + 4x_2 &=2\\ x_2 + x_3 &= 3 \end{aligned}
Solving Linear Systems
Practice: Solving a Homogeneous Linear System
Solve the following homogeneous system of linear equations. What type of solution is it?
2x1−x2+2x4=02x2+4x3−x4=0\begin{aligned} 2x_1-x_2+2x_4 &= 0\\ 2x_2 + 4x_3 − x_4 &= 0 \end{aligned}
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]
Find I0 in the following circuit.
Solving Linear Systems
Example: Solving a Linear System (Unique Solution)
Solve the system of linear equations:
2x+2y−2z=2x+2z−7=03y=−7−2z\begin{array}{ccccccl} 2x&+&2y&-&2z&=&2\\ x&+&2z&-&7&=&0\\ &&&&3y &=&-7-2z \end{array}
Solving Linear Systems
Example: Solving a Linear System (Infinitely Many Solutions)
Solve the following system of linear equations:
x+2y=4y+z=3z+w=1\begin{array}{c} x+2y&=&4\\ y+z&=&3\\ z+w&=&1 \end{array}
19.4F_Final_Builder_Ch_2.13_Algebra_SLE_$\tkco{eg22}$_$\key{Final}$_Builder_$\tkcth{2.13.}\tkcf{22}$_
Given the system of equations:
{2x1−4x2=−1−4x1+8x2=2 \left\{ \begin{array}{ll} 2x_1 - 4x_2 & = \quad -1 \\ -4x_1 + 8x_2 & = \quad 2 \end{array} \right.
Show that x⃗g=x⃗p+tx⃗h\vx_g = \vx_p + t \vx_h, where x⃗p=[−120]\vx_p = \colvec{-\frac12}{0} and x⃗h=[21]\vx_h = \colvec{2}{1} is a solution to the system when t=1,2,3t = 1, 2, 3, i.e. show that the vectors:
Find (if possible) conditions on aa and bb so that the following system has
(a) no solution
(b) one solution
Which of the following options is the general solution of the following system of linear equations?
x−2y+z=3x−y−z=0\begin{array}{l}x-2y+z=3\\x-y-z=0\end{array}
What can you say about the following systems of linear equations?
x+y−2z=22x+2y−6z=3\begin{array}{l}x+y-2z=2\\2x+2y-6z=3\end{array}
What can you say about the following systems of linear equations?
x+y−2z=−32x+2y−4z=−6\begin{array}{l}x+y-2z=-3\\2x+2y-4z=-6\end{array}
Matrix Multiplication
e.g. Find a 3 × 3 matrix M such that
 ⁣M[   0−2   0]=[−1   1−1] ⁣, ⁣M[001]=[   0−2   3] ⁣,and M[1300]=[−2   2−1]\!M\begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 0 \end{bmatrix}= \begin{bmatrix} -1\\\ \ \ 1\\-1 \end{bmatrix}\!, \hspace{0.7cm} \!M\begin{bmatrix} 0\\0\\1 \end{bmatrix}= \begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 3 \end{bmatrix}\!, \hspace{0.7cm} \text{\small{and}}\ M\begin{bmatrix} \frac{1}{3}\\0\\0 \end{bmatrix}= \begin{bmatrix} -2\\\ \ \ 2\\-1 \end{bmatrix}
System of Linear Equations
Find (if possible) conditions on aa and bb so that the following system has
(a) no solution
(b) one solution
Homogeneous System of Equations
Find basic solutions of a homogeneous system of linear equations with reduced row echelon form of:
[1203000150]\left[\begin{array}{cccc|c}1 & 2 & 0 & 3 & 0 \\0 & 0 & 1 & 5 & 0\end{array}\right]
Practice Question: Solving a System of Linear Equations (Unique Solution)
Practice Question: Solving a SLE--Unique Solution
Solve the following systems of linear equations.
𝑥−𝑧=5𝑥−𝑧=5
Find size of soln set to nonhom linear system
Find all solutions to the system of linear equations 5x+y+z=0x+2z=−12x+y−5z=1\begin{array}{rcl} 5x+y+z=0\\ x+2z=-1\\ 2x+y-5z=1 \end{array}.
What can you say about the following systems of linear equations?
x+y−2z=22x+2y−6z=3\begin{array}{l}x+y-2z=2\\2x+2y-6z=3\end{array}
Write the general solution (all solutions) to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l}x-y-3z=-4\\2x-3y+z=0\end{array}
Find (if possible) conditions on aa and bb so that the following system has
(a) no solution
(b) one solution
What can you say about the following systems of linear equations?
x+y−2z=−32x+2y−4z=−6\begin{array}{l}x+y-2z=-3\\2x+2y-4z=-6\end{array}
Which of the following options is the general solution of the following system of linear equations?
x−2y+z=3x−y−z=0\begin{array}{l}x-2y+z=3\\x-y-z=0\end{array}
Find all solutions to the system of linear equations 5x+y+z=0x+2z=−12x+y−5z=1\begin{array}{c} 5x+y+z=0\\ x+2z=-1\\ 2x+y-5z=1 \end{array}.
Practice Question: Solving a System of Linear Equations (Infinitely Many Solutions)
Practice Question: Solving a SLE
Solve the following system of linear equations: x+2y=4y+z=3z+w=1\begin{array}{rcl} x+2y&=&4\\ y+z&=&3\\ z+w&=&1 \end{array}.
Solve the homogeneous linear system
When the right hand side of every equation in a linear system is 0, we say that the system is homogeneous
Find the general solution of the following homogeneous linear system:
{3x+2y=0−6x+4y=0\left\{\begin{array}{rcl} 3x+2y&=&0\\ -6x+4y&=&0 \end{array} \right.
Solving a system of linear equations
Solve the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr} -2x_1+3x_2+13x_3&=&5\\ -x_1+2x_2+5x_3&=&1\\ 4x_1-14x_2-2x_3&=&14\\ \end{array}
Solving a system of linear equations
Solve the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr} -2x_1+3x_2+13x_3&=&5\\ -x_1+2x_2+5x_3&=&1\\ 4x_1-14x_2-2x_3&=&14\\ \end{array}
Solving a system of linear equations
Solve the following system of linear equations:
−2x1−4x2−6x3−19x4=−153x1+10x2−30x3+9x4=24x1+3x2−6x3+4x4=87x1+17x2+45x4=55\begin{array}{rcr} -2x_1-4x_2-6x_3-19x_4&=&-15\\ 3x_1+10x_2-30x_3+9x_4&=&24\\ x_1+3x_2-6x_3+4x_4&=&8\\ 7x_1+17x_2+45x_4&=&55 \end{array}
where we wish to solve for the quadruple (x1,x2,x3,x4)(x_1,x_2,x_3,x_4) of real numbers that satisfy this system
Practice: Solutions to a System of Equations
Practice: Solutions to a System of Equations
Given the system of linear equations x+y=0x−cy=1\begin{array}{c} x+y=0\\ x-cy=1 \end{array}, find the values of cc such that,
Solving a system of linear equations
Solve the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr} -2x_1+3x_2+13x_3&=&5\\ -x_1+2x_2+5x_3&=&1\\ 4x_1-14x_2-2x_3&=&14\\ \end{array}
where we wish to solve for the triple (x1,x2,x3)(x_1,x_2,x_3) of real numbers that satisfy this system
Practice: Solving a System of Equations
Practice: Solving a System of Equations
Solve the system of equations 2x+2y−2z=2x+2z−7=03y=−7−2z\begin{array}{rcl} 2x+2y-2z&=&2\\ x+2z-7&=&0\\ 3y&=&-7-2z \end{array}
Linear algebra
For which values of aaand bbdoes the system x+3y=2b3x−3ay=a\begin{array}{l}x+3y = 2b\\3x-3ay = a\end{array}have infinitely many solutions?
For which values of a and b does the system x+3y=2b3x−3ay=a\begin{array}{l}x+3y = 2b\\3x-3ay = a\end{array}have infinitely many solutions?
For which values of a and b does the system 2x−4y=a3x−ay=3b\begin{array}{l}2x-4y = a\\3x-ay = 3b\end{array}have no solutions?
Write the general solution to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l} x - y - 3z = -4\\ 2x - 3y + z = 0 \end{array}
System of Linear Equations
Write the general solution to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l} x - y - 3z = -4\\ 2x - 3y + z = 0 \end{array}
Solve the following linear system of equations.
2x+3y−z=7x−2y+5z=27y−11z=6\begin{array}{c} 2x + 3y - z = 7\\ x - 2y + 5z = 2\\ 7y - 11z = 6 \end{array}
Solving Linear Systems
Solve the following homogeneous system of linear equations. What type of solution is it?
2x1−x2+2x4=02x2+4x3−x4=0\begin{aligned} 2x_1-x_2+2x_4 &= 0\\ 2x_2 + 4x_3 − x_4 &= 0 \end{aligned}
Solving Linear Systems
Solve the following system of linear equations. What type of solution is it?
3x1+6x2=12x1+4x2=2x2+x3=3\begin{aligned} 3x_1 + 6x_2&= 1\\ 2x_1 + 4x_2 &=2\\ x_2 + x_3 &= 3 \end{aligned}
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}