Question

Prove that if the set of vectors {u1, u2, u3} is linearly dependent and u4 is any vector, then the set {u1, u2, u3, u4} is linearly dependent.

Answer 1

Answer with Step-by-step explanation:

We are given that the set of vectors${u_1,u_2,u_3}$ is lineraly dependent set .

We have to prove that the set ${u_1,u_2,u_3,u_4}$ is linearly dependent .

Linearly dependent vectors : If the vectors $u_1,u_2.u_3,u_4$

are linearly  dependent therefore the linear combination

$a_1u_1+a_2u_2+a_3u_3+a_4u_4=0$

Then ,there exit a scalar which is not equal to zero .

Let $a_1\neq0$ then the vector $u_1$ will be zero and remaining  other vectors are not zero.

Proof:

When $u_1,u_2,u_3$ are linearly dependent vectors therefore, linear combination of vectors of given set

$a_1u_1+a_2u_2+a_3u_3=0$

By definition of linearly dependent vector

There exist a scalar which is not equal to zero.

Suppose $a_1\neq 0$ then  $u_1=0$

The linear combination of the set ${u_1,u_2,u_3,u_4}$

$a_1u_1+a_2u_2+a_3u_3+a_4u_4=0$

When $a_1\neq0\; and\; u_1=0$

Therefore,the set ${u_1,u_2,u_3,u_4}$ is linearly dependent because it contain a vector which is zero.

Hence, proved .