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

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

The matrix \( A \) is given by:

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

The **left null space** of a matrix \( A \) consists of all vectors \( \mathbf{v} \) such that:

\[
\mathbf{v}^T A = 0
\]

This means we need to find the solution to the system:

\[
\mathbf{v}^T A = 0 \quad \text{or equivalently} \quad A^T \mathbf{v} = 0
\]

### Steps to find the left null space:
1. Transpose the matrix \( A \).
2. Solve the homogeneous system \( A^T \mathbf{v} = 0 \).

Let's compute the left null space.The left null space of the given matrix \( A \) is spanned by the vector:

\[
\mathbf{v} = \begin{pmatrix} 0.8347 \\ 0.0273 \\ -0.5500 \end{pmatrix}
\]

This vector forms a basis for the left null space of matrix \( A \).

Would you like further details or any clarifications?

### Related questions:
1. How do you calculate the null space of a matrix?
2. What is the difference between the left null space and right null space of a matrix?
3. How do you find the row space of a matrix?
4. What role does singular value decomposition (SVD) play in finding the null space?
5. Can the left null space be found without transposing the matrix?

### Tip:
The left null space provides insight into constraints on the rows of a matrix, helping to find dependencies between them.

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

A =
⎛
2 4 1 3 5
−1 0 7 2 1
3 6 2 4 8
⎞

Learn how to find the basis of the left null space of a matrix with a step-by-step solution. This problem involves key concepts of linear algebra, including matrix transposition and solving homogeneous systems, suitable for undergraduate students.

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