Question 1
Which theorem states that in a linearly dependent set, one vector is a linear combination of others?
Question 2
How can we prove that a set of vectors is linearly independent?
Question 3
What implies the presence of a zero vector in a set containing vectors?
Question 4
What can be said about any subset of a linearly independent set?
Question 5
What happens to the linear dependence of vectors if you include the zero vector in any set?
Question 6
What does the presence of zero rows in the reduced matrix form indicate?
Question 7
If a subset of vectors is removed from a linearly independent set, what remains true?
Question 8
Which property is true if vectors V1, V2, ..., Vn are linearly dependent?
Question 9
What does it mean for vectors V1, V2, ..., Vn to be linearly dependent?
Question 10
What does it indicate if the only solution to α1 V1 + α2 V2 + ... + αn Vn = 0 is all α's being zero?
Question 11
Given vectors V1 = (1, -2, 1), V2 = (2, 1, -1), V3 = (7, -4, 1), how can their linear dependence be checked?
Question 12
In the context of vector spaces, how are polynomials and matrices treated to check for linear dependence?
Question 13
Why are vectors (1, 1, 0), (0, 1, 1), (1, 0, 1) considered linearly independent?
Question 14
What condition makes a new set of vectors linearly dependent when adding one vector to an already dependent set?
Question 15
What is the trivial solution in the context of linear dependence for vectors V1, V2, V3?