August 8, 2025
Examples:
The black boxes and boxed elements are called pivots and their location tells us about the solution spaces of the corresponding systems.
Reminder: Pivots do not necessarily occur on the diagonal, and it is not the case that every row or every column of the matrix must have a pivot.
Goal: Determine what and how the number and location of pivots in a matrix in row echelon form tell us about the solution space.
We first saw pivots in our Day 4 notebook.
pivots are the leading non-zero entries in each row of a matrix in row echelon form.
We identified that the positions of our pivots determine the size of our solution space corresponding to a linear system.
Example: The locations of the pivots in the augmented coefficient matrices below tell us about the size of the corresponding solution spaces.
\[\overset{\text{Unique Solution}}{\left[\begin{array}{cc|c}\boxed{~1~} & 0 & -3\\ 0 & \boxed{~1~} & 2\end{array}\right]}~~~~\overset{\text{No Solutions}}{\left[\begin{array}{cc|c}\boxed{~1~} & 0 & -3\\ 0 & 0 & \boxed{~1~}\end{array}\right]}~~~~\overset{\text{Infinitely Many Solutions}}{\left[\begin{array}{cc|c}\boxed{~1~} & 0 & -3\\ 0 & 0 & 0\end{array}\right]}\]
Limitations on Number of Pivots: Recognizing come obvious limitations can reduce the amount of work we need to do if we are only interested in existence or particularly uniqueness of solutions to systems.
In any \(m\times n\) matrix, there can be at most one pivot per row.
Similarly, in any \(m\times n\) matrix, there can be at most one pivot per column.
Example: If an augmented coefficient matrix is \(4\times 7\) (so the coefficient matrix is \(4\times 6\)), then the maximum number of pivots in the augmented coefficient matrix is \(4\).
Limitations on Number of Pivots: Recognizing come obvious limitations can reduce the amount of work we need to do if we are only interested in existence or particularly uniqueness of solutions to systems.
In any \(m\times n\) matrix, there can be at most one pivot per row.
Similarly, in any \(m\times n\) matrix, there can be at most one pivot per column.
Example: If an augmented coefficient matrix is \(4\times 7\) (so the coefficient matrix is \(4\times 6\)), then the maximum number of pivots in the augmented coefficient matrix is \(4\). This means that not every column in the coefficient matrix can be a pivot column. For this reason, the corresponding system cannot have a unique solution – it is either an inconsistent system or it has infinitely many solutions.
In an augmented coefficient matrix, each column to the left of the augmentation line corresponds to a variable in the system.
\[\left[\begin{array}{cccc|c} \overset{\mathbf{x_1}}{a_{11}} & \overset{\mathbf{x_2}}{a_{12}} & \cdots & \overset{\mathbf{x_n}}{a_{1n}} & b_1\\ a_{21} & a_{22} & \cdots & a_{2n} & b_2\\ \vdots & \vdots & \ddots & a_{1n} & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m\end{array}\right]\]
In an augmented coefficient matrix, each column to the left of the augmentation line corresponds to a variable in the system.
\[\left[\begin{array}{cccc|c} \overset{\mathbf{x_1}}{a_{11}} & \overset{\mathbf{x_2}}{a_{12}} & \cdots & \overset{\mathbf{x_n}}{a_{1n}} & b_1\\ a_{21} & a_{22} & \cdots & a_{2n} & b_2\\ \vdots & \vdots & \ddots & a_{1n} & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m\end{array}\right]\]
If a variable corresponds to a pivot column, then that variable is a basic variable.
If a variable does not correspond to a pivot column, then that variable is a free variable.
Remark: Once the values for all free variables have been chosen, then the values of the basic variables are determined.
Example: For each of the augmented coefficient matrices below, determine the following:
\[\small{(A)~~\left[\begin{array}{ccc|c} 1 & 0 & 0 & 2\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 3\end{array}\right]}\] \[\small{(B)~~\left[\begin{array}{ccc|c} 1 & 2 & -1 & 3\\ 0 & 1 & 4 & -2\\ 0 & 0 & 0 & 0\end{array}\right]}~~~\small{(C)~~\left[\begin{array}{cc|c} 0 & 1 & 2\\ 1 & 0 & 3\end{array}\right]}\]
\[\small{(D)~~\left[\begin{array}{ccc|c} 1 & 2 & -1 & 0\\ 0 & 1 & 3 & 2\\ 1 & 3 & 2 & 7\end{array}\right]}\] \[\small{(E)~~\left[\begin{array}{cccc|c} 1 & 0 & -2 & 4 & 1\\ 0 & 1 & 3 & -1 & 0\end{array}\right]}\]
Linear Systems and Matrix Equations: Converting a linear system into an augmented coefficient matrix really converts the system to an equivalent matrix equation \(A\vec{x} = \vec{b}\) which we solve instead.
The solution to such a matrix equation is a vector. Because of this, it is often convenient to write solutions to systems of linear equations as vectors.
Example: The linear system \(\left\{\begin{array}{rcr} x_1 - 2x_2 + x_3 & = & -5\\ 2x_2 + 3x_3 & = & 9\end{array}\right.\) corresponds to the matrix equation \(\begin{bmatrix} 1 & -2 & 1\\ 0 & 2 & 3\end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ x_3\end{bmatrix} = \begin{bmatrix} -5\\ 9\end{bmatrix}\)
\[\overset{\text{Linear System}}{\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.}~~~~~~~~\overset{\text{Matrix Equation}}{\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \cdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn}\end{bmatrix}\vec{x} = \begin{bmatrix} b_1\\ b_2\\ \vdots\\ b_m\end{bmatrix}}\]
When a linear system has a solution with free variables, the solution vector to the corresponding matrix equation \(A\vec{x} = \vec{b}\) will not be completely determined
Example: Consider the augmented coefficient matrix \(\left[\begin{array}{ccc|c} 1 & -2 & 1 & -5\\ 0 & 2 & 3 & 9\end{array}\right]\), which has reduced row echelon form \(\left[\begin{array}{ccc|c} 1 & 0 & 4 & 4\\ 0 & 1 & 3/2 & 9/2\end{array}\right]\).
Example: Consider the augmented coefficient matrix \(\left[\begin{array}{ccc|c} 1 & -2 & 1 & -5\\ 0 & 2 & 3 & 9\end{array}\right]\), which has reduced row echelon form \(\left[\begin{array}{ccc|c} 1 & 0 & 4 & 4\\ 0 & 1 & 3/2 & 9/2\end{array}\right]\).
Example: Consider the augmented coefficient matrix \(\left[\begin{array}{ccc|c} 1 & -2 & 1 & -5\\ 0 & 2 & 3 & 9\end{array}\right]\), which has reduced row echelon form \(\left[\begin{array}{ccc|c} 1 & 0 & 4 & 4\\ 0 & 1 & 3/2 & 9/2\end{array}\right]\).
\[\begin{align} \vec{x} &= \begin{bmatrix} x_1\\ x_2\\ x_3\end{bmatrix} \end{align}\]
Example: Consider the augmented coefficient matrix \(\left[\begin{array}{ccc|c} 1 & -2 & 1 & -5\\ 0 & 2 & 3 & 9\end{array}\right]\), which has reduced row echelon form \(\left[\begin{array}{ccc|c} 1 & 0 & 4 & 4\\ 0 & 1 & 3/2 & 9/2\end{array}\right]\).
\[\begin{align} \vec{x} &= \begin{bmatrix} x_1\\ x_2\\ x_3\end{bmatrix} = \begin{bmatrix} 4 - 4x_3\\ \frac{9}{2} - \frac{3}{2}x_3\\ x_3\end{bmatrix} \end{align}\]
Example: Consider the augmented coefficient matrix \(\left[\begin{array}{ccc|c} 1 & -2 & 1 & -5\\ 0 & 2 & 3 & 9\end{array}\right]\), which has reduced row echelon form \(\left[\begin{array}{ccc|c} 1 & 0 & 4 & 4\\ 0 & 1 & 3/2 & 9/2\end{array}\right]\).
\[\begin{align} \vec{x} &= \begin{bmatrix} x_1\\ x_2\\ x_3\end{bmatrix} = \begin{bmatrix} 4 - 4x_3\\ \frac{9}{2} - \frac{3}{2}x_3\\ x_3\end{bmatrix}\\ &= \begin{bmatrix} 4\\ \frac{9}{2}\\ 0\end{bmatrix} + \begin{bmatrix}-4x_3\\ -\frac{3}{2}x_3\\ x_3\end{bmatrix} \end{align}\]
Example: Consider the augmented coefficient matrix \(\left[\begin{array}{ccc|c} 1 & -2 & 1 & -5\\ 0 & 2 & 3 & 9\end{array}\right]\), which has reduced row echelon form \(\left[\begin{array}{ccc|c} 1 & 0 & 4 & 4\\ 0 & 1 & 3/2 & 9/2\end{array}\right]\).
\[\begin{align} \vec{x} &= \begin{bmatrix} x_1\\ x_2\\ x_3\end{bmatrix} = \begin{bmatrix} 4 - 4x_3\\ \frac{9}{2} - \frac{3}{2}x_3\\ x_3\end{bmatrix}\\ &= \begin{bmatrix} 4\\ \frac{9}{2}\\ 0\end{bmatrix} + \begin{bmatrix}-4x_3\\ -\frac{3}{2}x_3\\ x_3\end{bmatrix} = \begin{bmatrix} 4\\ \frac{9}{2}\\ 0\end{bmatrix} + x_3\begin{bmatrix}-4\\ -\frac{3}{2}\\ 1\end{bmatrix} \end{align}\]
Example: Consider the augmented coefficient matrix \(\left[\begin{array}{ccc|c} 1 & -2 & 1 & -5\\ 0 & 2 & 3 & 9\end{array}\right]\), which has reduced row echelon form \(\left[\begin{array}{ccc|c} 1 & 0 & 4 & 4\\ 0 & 1 & 3/2 & 9/2\end{array}\right]\).
So the solution space here consists of all vectors of the form \(\vec{x} = \begin{bmatrix} 4\\ \frac{9}{2}\\ 0\end{bmatrix} + x_3\begin{bmatrix} -4\\ -\frac{3}{2}\\ 1\end{bmatrix}\).
Geometry: This collection forms a line in three-dimensional space, through the point \(\begin{bmatrix} 4\\ \frac{9}{2}\\ 0\end{bmatrix}\) and sloped in the direction of the vector \(\begin{bmatrix} -4\\ -\frac{3}{2}\\ 1\end{bmatrix}\).
Reward: Parametric vector form gives full insight into the structure of the solution set for the underlying system. The ability to interpret the solution set geometrically, as a line in this case, gives us greater understanding of the solution vectors.
As in our example, the number of free variables determines the shape of the solution space. Each free variable corresponds to a directional vector that solutions can travel along.
Write the corresponding matrix equation and the augmented coefficient matrix. Transform the matrix into its equivalent reduced row echelon form and identify the pivot columns and highlight any free variables. Write the solution of the system in parametric vector form and describe the geometry of the solution space.
Write the corresponding matrix equation and the augmented coefficient matrix. Transform the matrix into its equivalent reduced row echelon form and identify the pivot columns and highlight any free variables. Write the solution of the system in parametric vector form and describe the geometry of the solution space.
Write the corresponding matrix equation and the augmented coefficient matrix. Transform the matrix into its equivalent reduced row echelon form and identify the pivot columns and highlight any free variables. Write the solution of the system in parametric vector form and describe the geometry of the solution space.
Write the corresponding matrix equation and the augmented coefficient matrix. Transform the matrix into its equivalent reduced row echelon form and identify the pivot columns and highlight any free variables. Write the solution of the system in parametric vector form and describe the geometry of the solution space.
Write the corresponding matrix equation and the augmented coefficient matrix. Transform the matrix into its equivalent reduced row echelon form and identify the pivot columns and highlight any free variables. Write the solution of the system in parametric vector form and describe the geometry of the solution space.
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