Visualizing a column space as a plane in R3 | Vectors and spaces | Linear Algebra | Khan Academy

TL;DR

The column space of a matrix is the span of its column vectors, and finding a linearly independent basis for the column space can be done by putting the matrix in reduced row echelon form.

Transcript

In the last video, I started with this matrix right here, and right from the get go, we said the span of this matrix is just the span of the column vectors of it, and I just wrote it right there. But we weren't clear whether this was linearly independent, and if it's not linearly independent, it won't be a sufficient basis. And then we go off and w... Read More

Key Insights

• 👾 The column space of a matrix is the span of its column vectors, which represents all possible linear combinations of those vectors.
• 🤨 Finding a linearly independent basis for the column space can be done by putting the matrix in reduced row echelon form and identifying the pivot entries and associated variables.
• ✈️ The column space can be visualized as a plane in R3, with the basis vectors specifying points on the plane and any linear combination of the basis vectors generating a vector in the plane.

Questions & Answers

Q: What is the column space of a matrix?

The column space of a matrix is the span of its column vectors, which represents all possible linear combinations of those vectors.

Q: How can a linearly independent basis for the column space be found?

The matrix can be put into reduced row echelon form, and the pivot entries and associated variables can be used to determine a basis for the column space.

Q: How does the column space relate to the solution set of Ax=b?

The solution set of Ax=b is valid only if the coefficients in the expression 2x - y - z + 3x = 0 equal zero, which reiterates that the column space must be a linearly independent set.

Q: How can the column space be visualized?

The column space can be visualized as a plane in R3, with the basis vectors specifying points on the plane and any linear combination of the basis vectors generating a vector in the plane.

Summary & Key Takeaways

• The span of a matrix is the span of its column vectors, which determines the column space.

• To find a linearly independent basis for the column space, the matrix can be put into reduced row echelon form.

• The reduced row echelon form reveals the pivot entries and associated variables, which can be used to determine a basis for the column space.

• The column space can also be visualized as a plane in R3, with the basis vectors specifying points on the plane.