Linear Combinations and Span in Linear Algebra

The video tutorial introduces the concept of linear combinations in linear algebra, explaining how vectors can be combined using arbitrary constants. It provides concrete examples and visualizations to illustrate the idea, showing how any vector in R2 can be represented as a linear combination of two non-collinear vectors. The tutorial also covers the concept of span, demonstrating that the span of two vectors can cover the entire R2 space if they are not collinear. An algebraic proof is provided to reinforce the understanding of linear combinations and their applications.

What is a linear combination in the context of vectors?

• A sum of vectors scaled by arbitrary constants

• A product of vectors

• A vector multiplied by itself

• A vector divided by a constant

In the example given, what is the result of 3 times vector a plus minus 2 times vector b?

• Vector 0, 0

• Vector 3, 0

• Vector 1, 2

• Vector 0, 3

Why is the term 'linear' used in linear combinations?

• Because vectors are squared

• Because vectors are added and scaled, not multiplied by each other

• Because vectors are multiplied by each other

• Because vectors are divided by constants

What does the span of two vectors a and b represent?

• The set of all vectors that can be formed by multiplying a and b

• The set of all vectors that can be formed by adding and scaling a and b

• The set of all vectors that can be formed by dividing a and b

• The set of all vectors that can be formed by subtracting a and b

Can any two vectors always span R2?

• No, not all pairs of vectors can span R2

• No, only collinear vectors can span R2

• No, only orthogonal vectors can span R2

• Yes, any two vectors can span R2

What is the span of the zero vector?

• All vectors in Rn

• All vectors in R3

• Only the zero vector

• All vectors in R2

What is required for two vectors to span all of R2?

• They must be identical

• They must be parallel

• They must be orthogonal

• They must be collinear

What is the algebraic method used to prove that any vector in R2 can be represented by a linear combination of two vectors?

• Matrix multiplication

• Vector division

• Scalar addition

• System of equations

In the algebraic proof, what is the result of solving for c2 in terms of x1 and x2?

• c2 = x1 + x2

• c2 = 2x1 + 3x2

• c2 = 1/3 x2 - x1

• c2 = x1 - x2

What is the significance of the vectors i and j in the context of spanning R2?

• They are zero vectors

• They are vectors that cannot span R2

• They are collinear vectors

• They are unit vectors that form a basis for R2