Math Problem Statement
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
Solution
The matrix is given by:
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.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Left Null Space
Matrix Transposition
Homogeneous Systems
Formulas
Left Null Space: v^T A = 0
Homogeneous System: A^T v = 0
Theorems
Null Space Theorem
Suitable Grade Level
Undergraduate Level
Related Recommendation
Find a Basis for the Null Space of Matrix A
Finding the Basis for Solution Space of a Homogeneous System of Equations
Finding the Basis of the Null Space for a 3x5 Matrix Using Gaussian Elimination
Basis for Solution Space of Homogeneous System in Linear Algebra
Find the Null Space and Basis of a Matrix A