Notes on Nullspace, Dependence, and Independence in Linear Algebra
Notes on Linear Algebra Homework
1. Nullspace of a Matrix
Concept
The nullspace (or kernel) of a matrix ( A ) consists of all vectors ( \mathbf{x} ) such that ( A \mathbf{x} = \mathbf{0} ). It represents the solutions to the homogeneous equation associated with the linear transformation defined by ( A ).
Given Matrix
[ A = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix} ]
Finding the Nullspace
To find the nullspace, set up the equation:
[ A \mathbf{x} = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix} ]
From the third row, we see that ( x_3 = 0 ). The first two rows provide no additional constraints, allowing ( x_1 ) and ( x_2 ) to be any real numbers.
Conclusion
Thus, the nullspace of ( A ) can be expressed as:
[ \text{Null}(A) = \left{ \begin{pmatrix} x_1 \ x_2 \ 0 \end{pmatrix} : x_1, x_2 \in \mathbb{R} \right} ]
This is a plane in (\mathbb{R}^3).
2. Examples of Linear Dependence and Independence
a) Linearly Dependent Set in ( \mathbb{R}^3 )
Example: The vectors [ \mathbf{v_1} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}, \quad \mathbf{v_2} = \begin{pmatrix} 2 \ 4 \ 6 \end{pmatrix}, \quad \mathbf{v_3} = \begin{pmatrix} 3 \ 6 \ 9 \end{pmatrix} ] are linearly dependent because each vector is a scalar multiple of the others.
Thoughts
In linear algebra, a set of vectors is dependent if at least one of them can be expressed as a linear combination of the others. This is crucial in determining the basis of a vector space.
b) Linearly Independent Set in ( P_3(\mathbb{R}) )
Example: The polynomials [ p_1(x) = 1, \quad p_2(x) = x, \quad p_3(x) = x^2 ] form a linearly independent set in ( P_3(\mathbb{R}) ) since none of these polynomials can be expressed as a linear combination of the others.
Additional Information
Linear independence is fundamental in understanding the structure of vector spaces. A basis for a vector space is formed by a set of linearly independent vectors that span the space.
Homework Information
| Course | Assignment | Weight | Deadline |
|---|---|---|---|
| Math 241 | Homework-2 (2.5%) | Each question (1,2) is worth equal credit. | 9th August |
Final Thoughts
Understanding the concepts of nullspace, linear dependence, and independence are foundational in linear algebra, impacting various applications such as solving systems of linear equations and transforming geometric interpretations in vector spaces.
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