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

Summary

This video script explores the concept of linear combinations in linear algebra, demonstrating how vectors can be scaled and added together to form new vectors. It uses examples to illustrate the idea and discusses the span of vectors, showing that certain combinations can represent all vectors in R2, while others can only represent a line or a single point.

Takeaways

• 📚 A linear combination of vectors is a sum of vectors scaled by arbitrary constants, where each constant is a real number.
• 🔍 The concept of a linear combination is fundamental in linear algebra and involves adding vectors after scaling them by constants.
• 📈 An example of a linear combination is given by the vectors \( \mathbf{a} = (1, 2) \) and \( \mathbf{b} = (0, 3) \), where any linear combination can be expressed as \( c_1 \mathbf{a} + c_2 \mathbf{b} \).
• 🧩 The 0 vector is a special case in linear combinations, where any scalar multiple of the 0 vector results in the 0 vector itself.
• 🔄 The term 'linear' in 'linear combination' emphasizes that vectors are being scaled and added, not multiplied in a way that would introduce nonlinearity.
• 🌐 The set of all vectors that can be formed by taking linear combinations of a given set of vectors is called the span of those vectors.
• 📉 Two vectors can span all of R2 (the set of all two-dimensional vectors) if they are not collinear, meaning they do not lie on the same line.
• 🔢 The span of a single vector is a line in R2, as it represents all possible scalar multiples of that vector.
• 📐 The unit vectors \( \mathbf{i} = (1, 0) \) and \( \mathbf{j} = (0, 1) \) form a basis for R2, meaning they can represent any vector in R2 through linear combinations.
• 📝 The formal definition of the span of a set of vectors is the set of all vectors that can be formed by taking linear combinations of those vectors, where the coefficients are any real numbers.

Q & A

What is a linear combination in the context of linear algebra?

A linear combination in linear algebra refers to the sum of vectors where each vector is multiplied by a scalar constant, which can be any real number. It represents a way to form new vectors from existing ones by scaling and adding them together.

Can you provide an example of a linear combination using two vectors?

Sure, if we have two vectors, say vector a = (1, 2) and vector b = (0, 3), a linear combination of these vectors could be 3*a - 2*b, which would result in the vector (3*1, 3*2) - (2*0, 2*3) = (3, 6) - (0, 6) = (3, 0).

What is the zero vector in linear algebra?

The zero vector is a vector with all its components equal to zero. It's denoted by a bold 0 and is the additive identity in vector spaces, meaning that any vector added to the zero vector results in the original vector.

Why is the term 'linear' important in the phrase 'linear combination'?

The term 'linear' is important because it specifies that the operation involves only scalar multiplication and vector addition, which are linear operations. It excludes non-linear operations such as vector multiplication or exponentiation.

What does it mean for a set of vectors to span a space?

When a set of vectors spans a space, it means that any vector within that space can be represented as a linear combination of the vectors in the set. In other words, the set of vectors is sufficient to describe every point in the space.

How can two vectors span the entire R2 space?

Two vectors can span the entire R2 space if they are not collinear, meaning they do not lie on the same line. This allows for any point in the two-dimensional space to be reached by scaling and adding these two vectors.

What is the span of the zero vector?

The span of the zero vector is just the zero vector itself. No matter what scalar is multiplied with the zero vector, the result is still the zero vector, so it does not span any space other than the single point at the origin.

Can any vector in R2 be represented by a single non-zero vector?

No, a single non-zero vector in R2 can only represent all the points along the line defined by that vector when scaled. It cannot represent the entire R2 space, as it lacks the component to move in the direction perpendicular to the line.

What are the unit vectors i and j in the context of R2?

In the context of R2, the unit vectors i and j are the basis vectors that are orthogonal to each other. i is represented as (1, 0) and j as (0, 1). They are used to express any vector in R2 as a linear combination of these two orthogonal vectors.

How can you algebraically prove that two vectors can span R2?

You can algebraically prove that two vectors can span R2 by showing that for any arbitrary point (x1, x2) in R2, there exist scalars c1 and c2 such that c1*vector1 + c2*vector2 equals (x1, x2). This demonstrates that any point in R2 can be reached by a linear combination of the two vectors.