Question

Determine whether the following statements are true or false and provide a justification for each response. Note that you must provide justification to receive full credit for each part.

a) If the set of vectors v₁, v₂, ....vₙ is linearly dependent, then one vector is a scalar multiple of one of the vectors.
b) If v₁, v₂....v₁₀ are vectors in R ⁵ , then the set of vectors is linearly dependent.
c) If v₁,v₂...v₅ are vectors in R¹⁰ , then the set of vectors is linearly independent. ¹
d) Suppose we have a set of vectors v₁,v₂...vₙ and that v₂ is a scalar multiple of v₁ . Then the set is linearly dependent.
e) Suppose the set of vectors v₁,v₂.....vₙ is linearly independent and form the columns of the matrix A. If the matrix vector equation Ax=b is consistent, then there is exactly one solution for that equation.

Answer 1

a) If the set of vectors v₁, v₂, ....vₙ is linearly dependent, then one vector is a scalar multiple of one of the vectors. b) If v₁, v₂....v₁₀ are vectors in R⁵ , then the set of vectors is linearly dependent. c) If v₁,v₂...v₅ are vectors in R¹⁰ , then the set of vectors is linearly independent. d) Suppose we have a set of vectors v₁,v₂...vₙ and that v₂ is a scalar multiple of v₁ . Then the set is linearly dependent. e) Suppose the set of vectors v₁,v₂.....vₙ is linearly independent and form the columns of the matrix A. If the matrix vector equation Ax=b is consistent, then there is exactly one solution for that equation.

Step-by-step explanation:

a) True: If the set of vectors v₁, v₂, ....vₙ is linearly dependent, then one vector is a scalar multiple of one of the vectors. This means that one vector in the set can be expressed as a scalar multiple of another vector in the set, which indicates linear dependence.

b) False: If v₁, v₂....v₁₀ are vectors in R⁵, then the set of vectors is linearly independent. To determine whether a set of vectors is linearly dependent, we need to check if the only solution to the homogenous linear combination equation is the trivial solution (where all the coefficients are zero), which is not necessarily guaranteed when there are more than 5 vectors in R⁵.

c) False: If v₁, v₂...v₅ are vectors in R¹⁰, then the set of vectors is linearly dependent. Similar to part b, a set of vectors in R¹⁰ will be linearly dependent if there exists a non-trivial solution to the homogenous linear combination equation.

d) True: If we have a set of vectors v₁, v₂...vₙ and v₂ is a scalar multiple of v₁, then the set is linearly dependent. This can be proven by finding a non-trivial solution to the linear combination equation.

e) True: If the set of vectors v₁, v₂...vₙ is linearly independent and form the columns of the matrix A, if the matrix vector equation Ax=b is consistent, then there is exactly one solution for that equation. This is due to the fact that linearly independent vectors form a basis, and a consistent solution indicates that the vector b can be represented as a unique linear combination of the vectors in the set.