Printable Practice Problems: Column, Row, Null, and Left Null Space

📚 Topic Summary

The column space of a matrix $A$ is the span of its column vectors. The row space is the span of its row vectors. The null space (or kernel) of $A$ consists of all vectors $x$ such that $Ax = 0$. The left null space of $A$ is the null space of $A^T$, i.e., all vectors $y$ such that $A^Ty = 0$. Understanding these spaces is fundamental for solving linear systems and understanding the properties of linear transformations.

Here's a worksheet to help you practice these concepts:

🧠 Part A: Vocabulary

Match the term with its definition:

1. Column Space
2. Row Space
3. Null Space
4. Left Null Space
5. Span

Definitions:

1. The set of all linear combinations of a set of vectors.
2. The set of all vectors $x$ such that $Ax = 0$.
3. The span of the column vectors of a matrix.
4. The span of the row vectors of a matrix.
5. The set of all vectors $y$ such that $A^Ty = 0$.

✏️ Part B: Fill in the Blanks

The _______ of a matrix $A$ is the set of all linear combinations of its column vectors. The _______ of $A$ is the set of all $x$ such that $Ax = 0$. The _______ of $A$ is the null space of $A^T$. The row space is the _______ of the row vectors.

🤔 Part C: Critical Thinking

Explain in your own words why understanding the null space of a matrix is important for solving systems of linear equations. Provide an example.