Linear combinations and span | Vectors and spaces | Linear Algebra | Khan Academy

TL;DR

Linear combinations involve adding and scaling vectors, and the span of a set of vectors represents all the vectors that can be obtained through linear combinations of those vectors.

Transcript

One term you are going to hear a lot of in these videos, and in linear algebra in general, is the idea of a linear combination. And all a linear combination of vectors are, they're just a linear combination. Let me show you what that means. So let's say I have a couple of vectors, v1, v2, and it goes all the way to vn. And they're all in, you know,... Read More

Key Insights

• 👻 Linear combinations involve adding and scaling vectors, allowing for the creation of new vectors.
• 😫 The span of a set of vectors represents all the vectors that can be obtained through linear combinations of those vectors.
• 🪡 To span all of R2, two non-collinear vectors are needed.
• 💁 Orthogonal vectors form a basis for R2 and simplify the process of finding linear combinations.
• 0️⃣ The span of the zero vector only consists of the zero vector itself.

Questions & Answers

Q: What is a linear combination?

A linear combination involves adding up vectors and scaling them by arbitrary constants. It allows for the creation of new vectors through vector addition and scalar multiplication.

Q: How can linear combinations be represented mathematically?

Linear combinations can be represented as c1v1 + c2v2 + ... + cn*vn, where c1 to cn are real numbers and v1 to vn are vectors.

Q: What is the span of a set of vectors?

The span of a set of vectors represents all the vectors that can be obtained through linear combinations of those vectors.

Q: Can any vector in R2 be represented by a linear combination of two vectors?

No, only two non-collinear vectors can span all of R2. The set of vectors must be linearly independent to span the full space.

Q: How does the concept of linear combinations relate to the concept of span?

Linear combinations are used to determine the span of a set of vectors. The span represents all the vectors that can be created by combining and scaling the given vectors.

Q: What is the significance of orthogonal vectors in linear combinations?

Orthogonal vectors, such as the i and j unit vectors, form a basis for R2 and can represent any vector in R2 through linear combinations. Orthogonal vectors simplify the process of finding linear combinations.

Q: Can the span of the zero vector represent any vector?

No, the span of the zero vector only consists of the zero vector itself. Any linear combination of the zero vector will result in the zero vector.

Q: What is the relationship between linear combinations and the concept of basis?

Linear combinations are used to determine the basis of a vector space. A basis is a set of vectors that spans the entire space and is linearly independent.

Summary & Key Takeaways

• Linear combinations involve adding up vectors while scaling them by arbitrary constants.

• The span of a set of vectors represents all the vectors that can be obtained through linear combinations of those vectors.

• The span of two non-collinear vectors can represent all vectors in a given space, such as R2.