Problem 21 - Differential Equations and Linear Algebra

Question

For Problems 19-23, find the dimension of the null space of the given matrix \(A\). \(A=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]\)

Step-by-Step Solution

Answer

The dimension of the null space of the given matrix A is 2. This is found by performing Gaussian elimination on the matrix, finding the RREF, and counting the number of free variables.

Step 1: Write down the given matrix A

The given matrix A is: \[A = \left[\begin{array}{lll}0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 1 & 0\end{array}\right]\]

Step 2: Perform Gaussian elimination

To find the RREF, perform a Gaussian elimination on the matrix A, row swap the first and third rows: \[\left[\begin{array}{lll}0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{array}\right]\]

Step 3: Determine the free variables

Now that we have reached the RREF of A, we can determine the free variables by looking at the pivot columns. The pivots are in the second row, so the free variables are in the first and third columns (variables \(x_1\) and \(x_3\)).

Step 4: Find the dimension of the null space

There are two free variables, so the dimension of the null space is 2. Therefore, the null space of matrix A has dimension 2.

Key Concepts

Gaussian elimination

Gaussian elimination is a systematic method for transforming a matrix into its row echelon form. This method involves performing operations on the rows of the matrix to create zeros below the pivot elements (the first non-zero element in each row). The process helps simplify the matrix for easier analysis. It's like tidying up a room by placing everything in its right place.

To perform Gaussian elimination, you can apply three types of row operations:

• Swap rows
• Multiply a row by a non-zero scalar
• Add or subtract a multiple of one row from another row

Using these steps, the goal is to reshape the matrix into an upper triangular form. This means it has zero entries below a line of pivots. The final objective is achieving the reduced row echelon form (RREF), a more refined version of the echelon form. By applying these operations to matrix \( A \), you revamp it into a simpler structure, making it easier to uncover its properties, such as finding free variables and reducing the matrix further into RREF in the process.

free variables

In linear algebra, free variables are those that can assume any value in a system of linear equations and are not directly determined by the pivot positions in the matrix. They contribute directly to the solutions of the equations by adding flexibility to the system.

When a matrix is in reduced row echelon form (RREF), identifying free variables becomes easier. Look at each column of the RREF:

• If a column contains a pivot (a leading 1 in its row), it is associated with a basic variable.
• If a column lacks a pivot, then it corresponds to a free variable.

In the given matrix example, the free variables emerge from the columns that lack pivot positions. Once identified, these free variables help in determining the dimension of the null space of the matrix. In our case, variables \(x_1\) and \(x_3\) become free variables, leading directly to finding that the null space dimension is 2.

reduced row echelon form

Achieving the reduced row echelon form (RREF) is the final goal of transforming a matrix through Gaussian elimination and other row operations. An RREF is a matrix that not only makes computations clear but also neatly summarizes the solutions of the system of linear equations obtained from it.

Columns with pivots in an RREF signal the positions of the basic variables, whereas columns without pivots indicate free variables. Remember, the key properties of a matrix in RREF are:

• Leading entries (or pivots) are all 1 and appear in a staircase pattern from left to right.
• Each pivot is the only non-zero entry in its column.
• The pivot is always to the right of any pivots in the rows above it.
• Any rows consisting entirely of zeros are positioned at the bottom of the matrix.

In the given scenario, the RREF transformation neatly aligns the information about where pivots lie, which guides you towards determining free variables and, ultimately, solving for the null space dimension of the matrix.