Understanding Basis in R3

The video tutorial explains how to determine if a set of vectors forms a basis for R3. It covers the conditions for a basis, which include independence and spanning. The tutorial demonstrates two tests: the independence test, which involves solving a homogeneous vector equation, and the spanning test, which checks if vectors can form any vector in R3. The video evaluates three sets of vectors, showing detailed steps for each test and concluding which sets form a basis for R3.

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

• Spanning and orthogonality

• Independence and spanning

• Independence and orthogonality

• Orthogonality and linearity

What does it mean if a set of vectors is independent?

• The vectors form a closed loop

• The only solution to the homogeneous equation is the trivial solution

• The vectors are orthogonal

• The vectors are parallel

In the independence test, what does a row of zeros in the reduced row echelon form indicate?

• The system is overdetermined

• The system is independent

• The system is inconsistent

• The system is dependent

What is the purpose of the spanning test?

• To check if vectors are orthogonal

• To determine if vectors can form any vector in R3

• To find the magnitude of vectors

• To calculate the angle between vectors

What does it mean if a set of vectors spans R3?

• The vectors are orthogonal

• The vectors are parallel

• The vectors can form any vector in R3

• The vectors are linearly dependent

What conclusion can be drawn if a set of vectors is both independent and spans R3?

• The set forms a basis for R3

• The set is orthogonal

• The set is linearly dependent

• The set is redundant

In the second set of vectors, what does the independence test reveal?

• The set is orthogonal

• The set is parallel

• The set is independent

• The set is dependent

For the second set of vectors, what does the spanning test confirm?

• The set spans R3

• The set is orthogonal

• The set is parallel

• The set does not span R3

What is the result of the independence test for the third set of vectors?

• The set is dependent

• The set is independent

• The set is orthogonal

• The set is parallel

What does the spanning test indicate about the third set of vectors?

• The set does not span R3

• The set is orthogonal

• The set spans R3

• The set is parallel