Linear Algebra Quiz

What does it mean for a set of vectors to span R3?

• They are linearly dependent

• They are orthogonal to each other

• They are linearly independent

• They can construct any vector in R3

What is the condition for a set of vectors to be linearly independent?

• The vectors are in the same plane

• The vectors are orthogonal to each other

• The solution to the linear combination is non-zero

• The only solution to the linear combination is when all constants are zero

If a set of three vectors span R3, what can be said about their linear independence?

• They are always linearly independent

• They are always linearly dependent

• They can be either linearly independent or dependent

• Their linear independence cannot be determined

What is the result of a linear combination of the three vectors when a, b, and c are all zero?

• The result is always the zero vector

• The result is always a non-zero vector

• The result is undefined

• The result is a scalar multiple of one of the vectors

If a set of three vectors are linearly independent, what can be concluded about their span?

• They span R3

• Their span cannot be determined

• They do not span R3

• They span a lower-dimensional space

What is the condition for a set of vectors to be linearly dependent?

• The only solution to the linear combination is when all constants are zero

• The solution to the linear combination is non-zero

• The vectors are orthogonal to each other

• The vectors are in the same plane

If a set of three vectors do not span R3, what can be said about their linear independence?

• Their linear independence cannot be determined

• They can be either linearly independent or dependent

• They are always linearly dependent

• They are always linearly independent

What is the result of a linear combination of the three vectors when a, b, and c are all non-zero?

• The result is undefined

• The result is always a non-zero vector

• The result is a scalar multiple of one of the vectors

• The result is always the zero vector

If a set of three vectors are linearly dependent, what can be concluded about their span?

• Their span cannot be determined

• They span a lower-dimensional space

• They do not span R3

• They span R3

What is the condition for a set of vectors to be linearly independent?

• The vectors are in the same plane

• The only solution to the linear combination is when all constants are zero

• The vectors are orthogonal to each other

• The solution to the linear combination is non-zero