Problem 22
Question
Find the dimension of the null space of the given matrix \(A\). $$ A=\left[\begin{array}{rrr} 1 & -1 & 4 \\ 2 & 3 & -2 \\ 1 & 2 & -2 \end{array}\right] $$
Step-by-Step Solution
Verified Answer
The matrix A, after converting to row-echelon form, has 2 pivot columns and 1 free column. Therefore, the dimension of the null space of matrix A is 1.
1Step 1: Write down the given matrix A.
We have the matrix A as:
$$
A=\left[\begin{array}{rrr}
1 & -1 & 4 \\\
2 & 3 & -2 \\\
1 & 2 & -2
\end{array}\right]
$$
2Step 2: Convert matrix A into row-echelon form using elementary row operations.
Let's perform the elementary row operations to get the row-echelon form of matrix A:
1. Subtract the first row from the third row: \(R_3 = R_3 - R_1\)
2. Subtract twice the first row from the second row: \(R_2 = R_2 - 2R_1\)
The matrix A now becomes:
$$
\left[\begin{array}{rrr}
1 & -1 & 4 \\
0 & 5 & -10 \\
0 & 3 & -6
\end{array}\right]
$$
Now, we need to make the pivot in the second row as 1 by dividing the second row by 5, i.e., \(R_2 = \frac{1}{5}R_2\). Also, subtract 3/5 times the second row from the third row: \(R_3 = R_3 - \frac{3}{5}R_2\). The matrix A becomes:
$$
\left[\begin{array}{rrr}
1 & -1 & 4 \\
0 & 1 & -2 \\
0 & 0 & 0
\end{array}\right]
$$
Now, the matrix A is in row-echelon form.
3Step 3: Identify pivot columns and free columns in the row-echelon form of matrix A.
In the row-echelon form of A, we have two pivot columns (columns with leading 1's): column 1 and column 2. We also have one free column (column without leading 1's): column 3.
4Step 4: Calculate the dimension of the null space.
The dimension of the null space of matrix A is equal to the number of free columns. In our case, there is only one free column (column 3), so the dimension of the null space of matrix A is 1.
Key Concepts
Row-Echelon FormPivot ColumnsFree ColumnsDimension of Null Space
Row-Echelon Form
Transforming a matrix into its row-echelon form is an essential step in linear algebra. This form makes it easier to solve systems of equations or to find certain properties of the matrix.
To achieve row-echelon form, each row of the matrix needs to have a leading 1 (called a pivot) that is situated to the right of any leading 1 in the row above. All non-zero rows are above any rows of all zeros.
Typically, you will use row operations such as swapping rows, multiplying a row by a non-zero scalar, or adding a multiple of one row to another row.
This process helps in simplifying the matrix, enabling you to identify pivot and free columns effectively, which further assists in determining the null space of the matrix.
To achieve row-echelon form, each row of the matrix needs to have a leading 1 (called a pivot) that is situated to the right of any leading 1 in the row above. All non-zero rows are above any rows of all zeros.
Typically, you will use row operations such as swapping rows, multiplying a row by a non-zero scalar, or adding a multiple of one row to another row.
This process helps in simplifying the matrix, enabling you to identify pivot and free columns effectively, which further assists in determining the null space of the matrix.
Pivot Columns
Pivot columns are the columns in a matrix's row-echelon form that contain leading 1s after performing row operations. These columns are essential because they correspond to independent variables in the matrix or system of equations.
- Pivots clarify the structure of the matrix and help to express some columns as a combination of other columns.
- In our example, the matrix row-echelon form reveals that the first and second columns are pivot columns.
- Each pivot represents an axis about which the rest of the matrix data can be described.
Free Columns
Free columns in the row-echelon form of a matrix are the columns that lack a pivot. These columns often correspond to the free variables in a system of equations.
- Free columns allow for multiple solutions. In terms of the null space, they provide the directions in which solutions can vary freely.
- From our matrix, column 3 is classified as a free column because it does not have a leading 1.
Dimension of Null Space
The dimension of the null space of a matrix is determined by counting the number of its free columns. This calculation highlights how much freedom exists in the solutions of the homogeneous equation associated with the matrix.
In simpler terms, the dimension of the null space represents the number of parameters or degrees of freedom within solutions.
In simpler terms, the dimension of the null space represents the number of parameters or degrees of freedom within solutions.
- It's equivalent to the number of free columns, which are not controlled by pivot columns.
- For our matrix example, with one free column, the null space's dimension is 1.
Other exercises in this chapter
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