Problem 7 - Differential Equations and Linear Algebra

Question

Determine the nullity of \(A\) "by inspection" by appealing to the Rank-Nullity Theorem. Avoid computations. $$A=\left[\begin{array}{rrrrr}1 & 3 & -3 & 2 & 5 \\\\-4 & -12 & 12 & -8 & -20 \\\0 & 0 & 0 & 0 & 0 \\\1& 3 & -3 & 2 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

By inspecting the given matrix A, we can identify two linearly independent rows (row 1 and row 4). Therefore, the rank of matrix A is 2. The Rank-Nullity Theorem states that rank(A) + nullity(A) = number of columns in A. Since the number of columns in matrix A is 5, we can find the nullity of matrix A as follows: Nullity(A) = 5 - Rank(A) = 5 - 2 = 3. Thus, the nullity of matrix A is 3.

Step 1: Identify linearly independent rows

Inspect the given matrix A and notice that row 1 and row 4 are linearly independent rows. Observe that row 2 is -4 times row 1 and row 3 is a row of all zeros.

Step 2: Calculate the rank

Since row 1 and row 4 are linearly independent, the rank of matrix A is 2.

Step 3: Apply the Rank-Nullity Theorem

The Rank-Nullity theorem states that rank(A) + nullity(A) = number of columns in A. In this case, the number of columns in matrix A is 5. To find the nullity of matrix A, plug in the rank(A) value we have found: Nullity(A) = Number of columns in A - Rank(A) Nullity(A) = 5 - 2

Step 4: Calculate nullity

Calculate the nullity of matrix A: Nullity(A) = 3 The nullity of matrix A is 3.

Key Concepts

Linear Independence

Linear independence in matrices refers to rows or columns that cannot be written as a combination of others. When dealing with matrices, identifying linearly independent rows and columns is crucial because it directly affects the matrix's rank.
In the context of the given exercise, you first look at each row of matrix \( A \) to see if any row can be expressed as a combination of others.

• Row 1 and Row 4 in the matrix are linearly independent since neither can be expressed as a multiple or sum of other rows.
• Row 2 is not linearly independent because it is simply -4 times Row 1. Thus, it does not contribute to the rank.
• Row 3 consists of all zeros and automatically does not affect linear independence.

Understanding linear independence allows us to determine other properties of the matrix, notably the rank.

Matrix Rank

The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix. It gives us a sense of the "dimension" of the space that the matrix spans.
To determine the matrix rank by inspection, you simply need to count the number of linearly independent rows (or columns) in the matrix. With matrix \( A \), the steps go as follows:

• Identify the linearly independent rows, which we've found are rows 1 and 4.
• Since rows 1 and 4 are independent and there are no more independent rows, the rank of the matrix is 2.

Thus, the rank is an essential concept toward understanding the solution space of a matrix and is a key part of the Rank-Nullity Theorem.

Nullity of a Matrix

The nullity of a matrix is the dimension of the kernel (null space) of the matrix, which is the set of solutions to the equation \( A\mathbf{x} = 0 \). The null space consists of all the vectors \( \mathbf{x} \) that when multiplied by \( A \) result in the zero vector.
In the Rank-Nullity Theorem, nullity is defined in the context of the total number of columns in a matrix: - Rank(A) + Nullity(A) = Number of Columns(A)Given matrix \( A \), with a rank of 2 and 5 columns total, you can calculate the nullity as:

• Nullity(A) = Number of columns - Rank(A)
• Nullity(A) = 5 - 2 = 3

The nullity tells us about the number of free variables in the solution to \( A\mathbf{x} = 0 \), in this case, there are 3. Essentially, it gives you an understanding of the degrees of freedom present within the matrix's solution space.