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Practice Question 6: Solving Systems of Linear Equations

Related Topics

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

Geometric Interpretation of Linear Systems

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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
−x1+x2+x3=0-x_1+x_2+x_3=0
−x2+3x3=3-x_2+3x_3=3

b) x1−2x2+3x3=5x_1-2x_2+3x_3=5
−x1+2x2−3x3=−5-x_1+2x_2-3x_3=-5
x2+2x3=3x_2+2x_3=3

c) x1+x2+x3=1x_1+x_2+x_3=1
−x1−x2−x3=−1-x_1-x_2-x_3=-1
2x1+2x2+2x3=22x_1+2x_2+2x_3=2
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
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.
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}