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Solve the System of Linear Equations and Give the Vector Form for the General Solution
Solve the following system of linear equations and give the vector form for the general solution.
x_1 -x_3 -2x_5&=1 \\
x_2+3x_3-x_5 &=2 \\
2x_1 -2x_3 +x_4 -3x_5 &= 0
(The Ohio State University, linear algebra midterm exam problem)
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Midterm 1 problems and solutions
We solve the system by Gauss-Jordan elimination.
The augmented matrix of the system is given by
\[\left[\begin{array}{rrrrr|r}
1 & 0 & -1 & 0 &-2 & 1 \\
0 & 1 & 3 & 0 & -1 & 2 \\
2 & 0 & -2 & 1 & -3 & 0 \\
\end{array}\right].\] We apply the elementary row operations as follows.
\left[\begin{array}{rrrrr|r}
\end{array}\right] \xrightarrow{R_3-2R_1}
0 & 0 & 0 & 1 & 1 & -2 \\
\end{array}\right].
Then the last matrix is in reduced row echelon form.
The variables $x_1, x_2, x_4$ correspond to the leading $1$'s of the last matrix, hence they are dependent variables and the rest $x_3, x_5$ are free variables.
From the last matrix we obtain the general solution
x_1&=x_3+2x_5+1\\
x_2&=-3x_3+x_5+2\\
x_4&=-x_5-2.
The vector form for the general solution is obtained by substituting these into the vector $\mathbf{x}$.
\mathbf{x}&=\begin{bmatrix}
x_1 \\
x_5
\end{bmatrix}=\begin{bmatrix}
x_3+2x_5+1 \\
-3x_3+x_5+2 \\
-x_5-2 \\
\end{bmatrix}\\[10pt] &=x_3\begin{bmatrix}
1 \\
-3 \\
\end{bmatrix}+x_5\begin{bmatrix}
\end{bmatrix}
+\begin{bmatrix}
\end{bmatrix}.
Therefore, the vector form for the general solution is given by
\[\mathbf{x}=x_3\begin{bmatrix}
\end{bmatrix},\] where $x_3, x_5$ are free variables.
This is one of the midterm exam 1 problems of linear algebra (Math 2568) at the Ohio State University.
The following list is the problems and solutions/proofs of midterm exam 1 of linear algebra at the Ohio State University in Spring 2017.
Problem 1 and its solution: Possibilities for the solution set of a system of linear equations
Problem 2 and its solution (The current page): The vector form of the general solution of a system
Problem 3 and its solution: Matrix operations (transpose and inverse matrices)
Problem 4 and its solution: Linear combination
Problem 5 and its solution: Inverse matrix
Problem 6 and its solution: Nonsingular matrix satisfying a relation
Problem 7 and its solution: Solve a system by the inverse matrix
Problem 8 and its solution:A proof problem about nonsingular matrix
Click here if solved 42
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Express a Vector as a Linear Combination of Given Three Vectors Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 […]
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Linear Algebra Problems by Topics
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Introduction to Matrices
Elementary Row Operations
Gaussian-Jordan Elimination
Solutions of Systems of Linear Equations
Linear Combination and Linear Independence
Nonsingular Matrices
Inverse Matrices
Subspaces in $\R^n$
Bases and Dimension of Subspaces in $\R^n$
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Linearly Independency of General Vectors
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Linear Transformation from $\R^n$ to $\R^m$
Linear Transformation Between Vector Spaces
Orthogonal Bases
Determinants of Matrices
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Introduction to Eigenvalues and Eigenvectors
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Diagonalization of Matrices
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Dot Products and Length of Vectors
Eigenvalues and Eigenvectors of Linear Transformations
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