Coordinates with respect to a basis | Linear Algebra | Khan Academy

TL;DR

Vectors in a subspace can be represented as linear combinations of basis vectors, and their coordinates with respect to a basis can be used to specify their position within the subspace.

Transcript

Let's say I've got some subspace of Rn. Let's say V is a subspace of Rn. And let's say the set B-- I'll do it in blue-- let's say the set B is a basis for V, so it's got a bunch of vectors in it. And let's say it's got v1, v2, all the way to vk. So you can see we have k vectors, so v is a k-dimensional subspace. So that means that if I have some ve... Read More

Key Insights

• 😑 Vectors in a subspace can be expressed as linear combinations of basis vectors.
• 🫡 The coordinates with respect to a basis represent the weights or constants in the linear combination.
• #️⃣ The dimension of a subspace determines the number of coordinates required to specify a vector.

Questions & Answers

Q: What is a basis in linear algebra?

In linear algebra, a basis is a set of linearly independent vectors that span a subspace. Any vector in the subspace can be expressed as a linear combination of the basis vectors.

Q: How are coordinates related to a basis in linear algebra?

The coordinates of a vector with respect to a basis represent the weights or constants in the linear combination of basis vectors required to obtain that vector. The coordinates specify the position of the vector within the subspace.

Q: Can different bases be used to represent the same vector?

Yes, different bases can be used to represent the same vector. The coordinates of the vector will differ depending on the chosen basis, but the vector itself remains unchanged.

Q: What is the relationship between dimensions and coordinates?

The dimension of a subspace determines the number of coordinates required to uniquely specify a vector within that subspace. A k-dimensional subspace will have k coordinates associated with each vector in the subspace.

Summary & Key Takeaways

• A subspace of Rn can be represented using a basis set of vectors, where the number of basis vectors determines the dimension of the subspace.

• Any vector in the subspace can be expressed as a linear combination of the basis vectors, with the constants in the combination representing the coordinates of the vector with respect to the basis.

• The dimension of the subspace determines the number of coordinates required to uniquely specify a vector within the subspace.

• The standard basis vectors in Rn can also be used to represent vectors, and their weights in the linear combination can be considered as the standard coordinates.