Notes on Nullspace, Dependence, and Independence in Linear Algebra

Notes on Linear Algebra Homework

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1. Nullspace of a Matrix

Concept

The nullspace (or kernel) of a matrix AA consists of all vectors x\mathbf{x} such that Ax=0A \mathbf{x} = \mathbf{0}. It represents the solutions to the homogeneous equation associated with the linear transformation defined by AA.

Given Matrix

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. **Reference:** ::urlcard[Null space 3: Relation to linear independence (video) - Khan Academy]{src="https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/null-column-space/v/null-space-3-relation-to-linear-independence"} ::urlcard[[PDF] Math 2331 – Linear Algebra - 4.2 Null Spaces, Column Spaces ...]{src="https://www.math.uh.edu/~jiwenhe/math2331/lectures/sec4_2.pdf"} ::urlcard[[PDF] Math 241 Homework]{src="https://www.math.cmu.edu/~mradclif/teaching/241F18/Homework/Oct18.pdf"}