📚 Topic Summary
The null space of a matrix $A$ consists of all vectors $x$ such that $Ax = 0$. Finding a basis for the null space involves solving the homogeneous equation $Ax = 0$ and expressing the solution in parametric vector form. The vectors in the parametric form then form a basis for the null space.
In essence, you're finding all the vectors that, when multiplied by your matrix, result in the zero vector. The 'basis' is the smallest set of vectors needed to describe all such vectors.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Null Space |
A. A set of linearly independent vectors that span the null space. |
| 2. Basis |
B. The solution set of the equation $Ax = 0$. |
| 3. Matrix |
C. An array of numbers arranged in rows and columns. |
| 4. Vector |
D. A quantity with both magnitude and direction, often represented as a column of numbers. |
| 5. Linear Independence |
E. A set of vectors where none can be written as a linear combination of the others. |
(Match the numbers 1-5 with the letters A-E)
✍️ Part B: Fill in the Blanks
The null space of a matrix $A$ is the set of all vectors $x$ that satisfy the equation $Ax = $ ________. To find a basis for the null space, we first solve the equation $Ax = 0$ and write the solution in ________ vector form. The vectors in the parametric form then form a ________ for the null space. These vectors must be ________ independent.
🤔 Part C: Critical Thinking
Explain in your own words why finding a basis for the null space is useful in understanding the properties of a linear transformation.