Understanding Basis for R3

The video tutorial explains how to determine if a set of vectors forms a basis for R3. It covers the concepts of linear independence and dependence, using vector equations and augmented matrices to analyze three sets of vectors. The first set is found to be linearly dependent, while the second and third sets are linearly independent and form a basis for R3. The tutorial provides a step-by-step approach to identifying trivial and non-trivial solutions, emphasizing the importance of having a linearly independent set to span R3.

What are the two main conditions for a set of vectors to form a basis for R3?

• The set must be linearly dependent and span R3.

• The set must be linearly dependent and not span R3.

• The set must be linearly independent and span R3.

• The set must be linearly independent and not span R3.

How do we determine if a set of vectors is linearly independent?

• By verifying the vectors have the same magnitude.

• By ensuring the vectors are orthogonal.

• By solving the vector equation and checking for only the trivial solution.

• By checking if the vectors are parallel.

What does a row of zeros in the reduced row echelon form indicate about the set of vectors?

• The vectors do not span R3.

• The vectors are linearly independent.

• The vectors form a basis for R3.

• The vectors are linearly dependent.

In the first set of vectors, what does the presence of a free variable indicate?

• The vectors are linearly independent.

• The vectors are linearly dependent.

• The vectors form a basis for R3.

• The vectors are orthogonal.

What conclusion can be drawn if the only solution to the vector equation is the trivial solution?

• The set of vectors is orthogonal.

• The set of vectors does not form a basis for R3.

• The set of vectors is linearly independent.

• The set of vectors is linearly dependent.

What does the absence of a row of zeros in the reduced row echelon form suggest about the second set of vectors?

• The vectors are parallel.

• The vectors do not form a basis for R3.

• The vectors are linearly independent.

• The vectors are linearly dependent.

For the second set of vectors, what does each row in the reduced row echelon form indicate about the variables?

• Each variable is dependent on the others.

• Each variable equals zero, indicating linear independence.

• Each variable is greater than zero.

• Each variable is a free variable.

What is confirmed about the third set of vectors based on the reduced row echelon form?

• They are linearly independent and form a basis for R3.

• They do not form a basis for R3.

• They are orthogonal.

• They are linearly dependent.

Why do the second and third sets of vectors form a basis for R3?

• Because they are linearly dependent.

• Because they are linearly independent and span R3.

• Because they are orthogonal.

• Because they have the same magnitude.

What is the significance of having three vectors in R3?

• They automatically form a basis for R3.

• They can only form a basis if they are linearly independent.

• They are always linearly dependent.

• They cannot span R3.