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Solving 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)

Reduced Row Echelon Form (RREF)

6 Activities

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.
Then, determine which of the following is the correct geometrical interpretation of the solution.
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.
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 Reduced Row Echelon Form (RREF) Questions:
Find all the values of cc so that the following system of linear equations corresponding to the augmented matrix has a unique solution.
[22410c112c∣221]\left[\begin{array}{l} 2 & 2 & 4 \\ 1 &0 & c \\ 1 & 1 & 2c \\ \end{array}\right.\left|\begin{array}{l} 2\\2\\1\end{array}\right]
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].
Example: RREF
Example:
Which of the following are in reduced row echelon form?
a.) [100201000010]\begin{bmatrix} 1 &0 &0& 2\\ 0& 1 & 0 &0\\ 0&0&1&0\\ \end{bmatrix}
The reduced 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]_.
a) What is the rank of matrix A
How many non-pivot (non-leading) columns can the reduced row echelon form of a non-zero 2×32\times3 matrix have?
Rank of a Matrix
Practice: Rank
In the reduced row echelon form of the coefficient matrix AA that corresponds to a system of linear equations with 6 equations in 9 unknowns, what is the maximum number of leading 1s?
Practice: Row Echelon Matrix
Which of the following correspond with the RREF of the matrix
Find the first row of the row-reduced echelon form of the matrix
A=[123−1−10013]A=\begin{bmatrix}1&2&3\\-1&-1&0\\0&1&3\end{bmatrix}
Which of the following is in RREF (reduced row echelon form)?
A=[1000001001000000]B=[1−100001−100000000]A=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad B=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}
C=[1−1000010000−10000]D=[1100011000010000]C=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad D=\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}
Let the following augmented matrix
[10−5−3017000(k+2)(k−4)(k−4)]\left[\begin{array}{ccc|c} 1 & 0 & -5 & -3\\ 0 & 1 & 7 & 0\\ 0 & 0 & (k+2)(k-4) & (k-4) \end{array} \right]
For what value(s) of kkdoes the system have no solution?
[10−5−3017000(k+2)(k−4)(k−4)] \left[\begin{array}{ccc|c} 1 & 0 & -5 & -3\\ 0 & 1 & 7 & 0\\ 0 & 0 & (k+2)(k-4) & (k-4) \end{array} \right]
For what value(s) of kkdoes the system have a one-parameter family of solutions (infinitely many solutions)?
[10−5−3017000(k+2)(k−4)(k−4)] \left[\begin{array}{ccc|c} 1 & 0 & -5 & -3\\ 0 & 1 & 7 & 0\\ 0 & 0 & (k+2)(k-4) & (k-4) \end{array} \right]
For what value(s) of kkdoes the system have a unique solution?
Which of the following matrices is/are in row-reduced echelon form?
a.)[101021000]\begin{bmatrix}1&0&1\\ 0&2&1\\0&0&0\end{bmatrix}
b.)[012001]\begin{bmatrix}0&1&2\\ 0&0&1\\\end{bmatrix}
How many non-pivot (non-leading) columns can the reduced row echelon form of a non-zero 2×32\times3 matrix have?
Find all values of aa and bb so that the system of linear equations corresponding to the following augmented matrix has a three-parameter family of solutions.
[1000000a000−1000b00]\left[ \begin{array}{rrrrr|r} 1&0&0&0&0&0\\ 0&a&0&0&0&-1\\ 0&0&0&b&0&0 \end{array} \right]
Writing the solution from RREF
Consider the linear system with unknowns x1,x2,x3x_1,x_2,x_3whose augmented matrix in reduced row echelon form is:
[1−30000100000]\left[ \begin{array}{rrr|r} 1&-3&0&0\\ 0&0&1&0\\ 0&0&0&0 \end{array} \right]
Write the general solution
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.
Find all the values of cc so that the following system of linear equations corresponding to the augmented matrix has a unique solution.
[22410c112c∣221]\left[\begin{array}{l} 2 & 2 & 4 \\ 1 &0 & c \\ 1 & 1 & 2c \\ \end{array}\right.\left|\begin{array}{l} 2\\2\\1\end{array}\right]
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].
Find all the values of kk such that the system of linear equations corresponding to 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]has infinitely many solutions.
Find all the values of kk such that the system of linear equations corresponding to 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] has a unique solution.
Which of the following is in RREF (reduced row echelon form)?
A=[1000001001000000]B=[1−100001−100000000]A=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad B=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}
C=[1−1000010000−10000]D=[1100011000010000]C=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad D=\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}
Reduced Row Echelon Form (RREF)
Which of the following is in RREF (reduced row echelon form)?
A=[1000001001000000]B=[1−100001−100000000]A=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad B=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}
C=[1−1000010000−10000]D=[1100011000010000]C=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad D=\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}
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.
Reduced Row Echelon Form
Which of the following matrices are in row-reduced echelon form? (select all that apply)