# Question: guided proof prove that if w is orthogonal to...

###### Question details

**Guided Proof :**

Prove that if **w** is orthogonal to each vector in
*S* = {**v**1, **v**2, . . . ,
**v***n*}, then **w** is
orthogonal to every linear combination of vectors in *S*.
Getting Started: To prove that **w** is orthogonal to
every linear combination of vectors in *S*, you need to show
that their inner product is 0.

(i) Write **v** as a linear combination of vectors,
with arbitrary scalars *c*1, . . . , *cn*, in
*S*.

(ii) Form the inner product of **w** and
**v**.

(iii) Use the properties of inner products to rewrite the inner
product _**w**, **v**_ as a linear
combination of the inner products _**w**,
**v***i*_, *i* = 1, . . . ,
*n*.

(iv) Use the fact that **w** is orthogonal to each
vector in *S* to lead to the conclusion that
**w** is orthogonal to **v**.