Find the basis of Null space of the given matrix A.

A =


2 4 1 3 5
−1 0 7 2 1
3 6 2 4 8



We are tasked with finding the basis for the null space of the given matrix \( A \), which is a \( 3 \times 5 \) matrix:

\[
A = 
\begin{pmatrix}
2 & 4 & 1 & 3 & 5 \\
-1 & 0 & 7 & 2 & 1 \\
3 & 6 & 2 & 4 & 8
\end{pmatrix}
\]

### Steps to Find the Basis of the Null Space:

1. **Set up the system of linear equations**:
   To find the null space, we solve \( A \mathbf{x} = \mathbf{0} \), where \( \mathbf{x} = \begin{pmatrix} x_1 & x_2 & x_3 & x_4 & x_5 \end{pmatrix}^T \). This gives the following system of equations:
   \[
   2x_1 + 4x_2 + x_3 + 3x_4 + 5x_5 = 0
   \]
   \[
   -x_1 + 0x_2 + 7x_3 + 2x_4 + x_5 = 0
   \]
   \[
   3x_1 + 6x_2 + 2x_3 + 4x_4 + 8x_5 = 0
   \]

2. **Write the augmented matrix**:
   The augmented matrix of the system \( A \mathbf{x} = \mathbf{0} \) is:
   \[
   \begin{pmatrix}
   2 & 4 & 1 & 3 & 5 & | & 0 \\
   -1 & 0 & 7 & 2 & 1 & | & 0 \\
   3 & 6 & 2 & 4 & 8 & | & 0
   \end{pmatrix}
   \]

3. **Reduce the augmented matrix to row echelon form**:
   We will now use Gaussian elimination to reduce this matrix to row echelon form.

I'll perform these steps now and provide the result.The basis of the null space of the given matrix \( A \) is spanned by the following two vectors:

\[
\mathbf{v_1} = \begin{pmatrix} 9 \\ -\frac{11}{2} \\ 1 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{v_2} = \begin{pmatrix} -6 \\ 2 \\ -1 \\ 0 \\ 1 \end{pmatrix}
\]

Thus, the null space consists of all linear combinations of these two vectors.

Would you like more details on how these vectors were derived or any further explanation?

Here are some related questions to explore:
1. What is the rank of matrix \( A \)?
2. How is the dimension of the null space related to the rank of \( A \)?
3. What would the null space look like if the matrix were square and full-rank?
4. How does the null space help in solving systems of linear equations?
5. How does Gaussian elimination relate to finding the null space?

**Tip**: The number of vectors in the null space's basis is called the nullity of the matrix, and it is related to the dimension of the matrix.

Find the basis of Null space of the given matrix A.

This page provides a step-by-step solution for finding the basis of the null space of a 3x5 matrix using Gaussian elimination. Learn how to set up the system of linear equations, form the augmented matrix, reduce it to row echelon form, and derive the vectors that span the null space. Suitable for undergraduate students studying linear algebra.

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