Visualizations of left nullspace and rowspace | Linear Algebra | Khan Academy

TL;DR

This video explains the subspaces associated with a matrix, including its null space, column space, left null space, and row space, and demonstrates their visual representation in relation to each other.

Transcript

In the last video I had this 2 by 3 matrix A right here, and we figured out all of the subspaces that are associated with this matrix. We figured out its null space, its column space, we figured out the null space and column space of its transpose, which you could also call the left null space, and the row space, or what's essentially the space spa... Read More

Key Insights

• 👾 The column space and left null space of a matrix are both subspaces of R2.
• 👾 The null space and row space of a matrix are subspaces of R3.
• 👾 The null space is a plane in R3, while the row space is a line in R3.
• 👾 The row space is orthogonal to the null space, and the column space is orthogonal to the left null space.
• 👾 The row space and null space are known as orthogonal complements of each other.

Questions & Answers

Q: What are the four subspaces associated with a matrix?

The four subspaces associated with a matrix are the null space, column space, left null space, and row space.

Q: What is the column space?

The column space is the span of the column vectors of the matrix. It represents all possible linear combinations of the column vectors.

Q: How is the left null space related to the null space of the transpose?

The left null space is equivalent to the null space of the transpose. Both represent the vectors that, when multiplied with the matrix (or its transpose), result in the zero vector.

Q: What is the visual representation of the row space?

The row space is represented by a line in three-dimensional space. It is orthogonal to the null space and represents all vectors that are perpendicular to the null space.

Summary & Key Takeaways

• The video discusses the subspaces associated with a given matrix, including its null space, column space, left null space, and row space.

• The column space is the span of the column vectors of the matrix.

• The left null space is the span of the row vectors of the matrix.