Understanding Linear Dependence of Vectors

The video tutorial explains how to determine the value of k that makes a set of vectors linearly dependent. It begins by introducing vectors u, v, and w, and discusses the concepts of linear dependence and independence. The tutorial then sets up a system of equations to express vector w as a linear combination of vectors u and v. By solving these equations, the values of the scalars are found, leading to the calculation of k. The tutorial concludes with a verification of the solution.

What is the third component of vector w?

• A sum of u and v

• A variable k

• A known constant

• A product of u and v

What condition makes a set of vectors linearly dependent?

• All vectors are scalar multiples of each other

• At least one vector is a linear combination of others

• All vectors have the same magnitude

• All vectors are orthogonal

Why are vectors u and v considered linearly independent?

• They have the same direction

• They are scalar multiples of each other

• They are not scalar multiples of each other

• They have the same magnitude

What is the first equation in the system of equations?

• c1 - c2 = 1

• c1 + c2 = 0

• 9c1 + 3c2 = k

• 4c1 - 7c2 = 2

How many unknowns are in the system of equations?

• Two

• Four

• One

• Three

What is the value of c1 after solving the system?

• 0

• 1

• 2

• 3

What is the value of c2 after solving the system?

• 1/7

• 2/7

• 3/7

• 4/7

What is the value of k that makes the vectors linearly dependent?

• 67/7

• 68/7

• 69/7

• 70/7

How is vector w expressed as a linear combination of vectors u and v?

• w = 3u + 4v

• w = (1/2)u + v

• w = 2u + 3v

• w = u + (2/7)v

What should you verify to ensure the linear combination is correct?

• The orthogonality of vectors

• The equality of the linear combination

• The direction of vectors

• The sum of magnitudes