Null space 2: Calculating the null space of a matrix | Linear Algebra | Khan Academy

TL;DR

The null space of a matrix A is the set of all vectors that, when multiplied by A, result in the zero vector.

Transcript

In the last video, I spoke somewhat theoretically about what a null space is and we showed that it is a valid subspace. But in this video let's actually calculate the null space for a matrix. In this case, we'll calculate the null space of matrix A. So null space is literally just the set of all the vectors that, when I multiply A times any of thos... Read More

Key Insights

• 😫 The null space of a matrix is the set of vectors that, when multiplied by the matrix, result in the zero vector.
• 👾 Calculating the null space involves solving the system of equations Ax = 0.
• 👾 The null space can be represented as the span of the vectors that form a linear combination of the solutions.

Questions & Answers

Q: What is the null space of a matrix?

The null space of a matrix A is the set of vectors that, when multiplied by A, result in the zero vector.

Q: How can the null space be calculated?

To calculate the null space, the system of equations Ax = 0 is solved.

Q: What is the relationship between the null space and row echelon form?

The null space of a matrix A is equal to the null space of its row echelon form.

Q: How can the null space be represented?

The null space can be represented as the span of the vectors that form a linear combination of the solutions.

Summary & Key Takeaways

• The null space of a matrix A is the set of vectors that, when multiplied by A, produce the zero vector.

• To find the null space, the system of equations Ax = 0 is solved.

• The null space can be represented as the span of the vectors that form a linear combination of the solutions.