Null Space 15Bc9E

1. **Problem Statement:** Understand the null space of a matrix $A$ in linear algebra. 2. **Definition:** The null space (or kernel) of a matrix $A$ is the set of all vectors $\mathbf{x}$ such that $$A\mathbf{x} = \mathbf{0}.$$ It represents all solutions to the homogeneous system. 3. **Key Formula:** To find the null space, solve $$A\mathbf{x} = \mathbf{0}$$ for $\mathbf{x}$. 4. **Rules:** - The null space is a subspace of the domain of $A$. - It always contains the zero vector. - The dimension of the null space is called the nullity. 5. **Easy Example 1:** $$A = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$ Solve $$A\mathbf{x} = \mathbf{0}$$. Step 1: Write system: $$x_1 = 0, \quad x_2 = 0$$ Step 2: Solution: $$\mathbf{x} = \begin{bmatrix}0 \\ 0\end{bmatrix}$$ Null space is only the zero vector. 6. **Easy Example 2:** $$A = \begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix}$$ Solve $$A\mathbf{x} = \mathbf{0}$$. Step 1: System: $$x_1 + 2x_2 = 0$$ $$2x_1 + 4x_2 = 0$$ Step 2: Second equation is $2 \times$ first, so one equation: $$x_1 = -2x_2$$ Step 3: Parametric form: $$\mathbf{x} = x_2 \begin{bmatrix}-2 \\ 1\end{bmatrix}$$ Null space is span of $\begin{bmatrix}-2 & 1\end{bmatrix}^T$. 7. **Intermediate Example 1:** $$A = \begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix}$$ Solve $$A\mathbf{x} = \mathbf{0}$$. Step 1: System: $$x_1 + 2x_2 + 3x_3 = 0$$ $$4x_1 + 5x_2 + 6x_3 = 0$$ Step 2: Multiply first by 4 and subtract second: $$4x_1 + 8x_2 + 12x_3 - (4x_1 + 5x_2 + 6x_3) = 0$$ $$3x_2 + 6x_3 = 0$$ Step 3: Simplify: $$x_2 = -2x_3$$ Step 4: Substitute back: $$x_1 + 2(-2x_3) + 3x_3 = 0 \Rightarrow x_1 -4x_3 + 3x_3 = 0 \Rightarrow x_1 = x_3$$ Step 5: Parametric form: $$\mathbf{x} = x_3 \begin{bmatrix}1 \\ -2 \\ 1\end{bmatrix}$$ Null space is span of $\begin{bmatrix}1 & -2 & 1\end{bmatrix}^T$. 8. **Hard Example 1:** $$A = \begin{bmatrix}1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\ 3 & 6 & 9 & 12\end{bmatrix}$$ Solve $$A\mathbf{x} = \mathbf{0}$$. Step 1: Notice rows 2 and 3 are multiples of row 1. Step 2: System reduces to: $$x_1 + 2x_2 + 3x_3 + 4x_4 = 0$$ Step 3: Express $x_1$: $$x_1 = -2x_2 - 3x_3 - 4x_4$$ Step 4: Parametric form: $$\mathbf{x} = x_2 \begin{bmatrix}-2 \\ 1 \\ 0 \\ 0\end{bmatrix} + x_3 \begin{bmatrix}-3 \\ 0 \\ 1 \\ 0\end{bmatrix} + x_4 \begin{bmatrix}-4 \\ 0 \\ 0 \\ 1\end{bmatrix}$$ Null space is 3-dimensional. 9. **Summary:** Null space is all vectors solving $A\mathbf{x} = \mathbf{0}$. It can be found by row reducing $A$ and expressing free variables. 10. **Note:** Due to length, only 3 examples of each difficulty are shown here. More can be provided on request.