Prove that if α 1 , α 2 , a n d α 3 are nonzero scalars and { u 1 , u 2 , u 3 } is linearly independent set of vectors, then { α 1 u 1 , α 2 u 2 , α 3 u 3 } is also a set of linearly independent vectors.

Question

Prove that if       α    1    ,      α    2    ,     a  n  d         α    3   are nonzero scalars and   {      u    1    ,      u    2    ,      u    3    } is linearly independent set of vectors, then  {      α    1        u    1    ,      α    2        u    2    ,      α    3        u    3    } is also a set of linearly independent vectors.

Paityn Arroyo · Accepted Answer

A vector is a direction with magnitude. A scalar is a value with only magnitude and no direction. If u1, u2 and u3 are independent vectors, then multiplying them by scalar values a1, a2 and a3 will only change the magnitude of the vector, and not remove the direction. E.g. if John rides his bike 2m east (a vector) one day, and doubles that distance the next day (so you&#039;re multiplying the magnitude by a vector), he rides his bike 4m east the second day. It doesn&#039;t mean he rides his bike 4 metres in no direction because that doesn&#039;t make sense.

Prove that if a_1, a_2, and a_3 are nonzero scalars and {u_1,u_2,u_3} is linearly independent set of vectors, then {a_1 u_1, a_2 u_2, a_3 u_3} is also a set of linearly independent vectors.

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