Proving the Basis Property for a Set in a Vector Space with a Nonzero Scalar

[trojansc82]

Homework Statement

Let c be a real scalar not equal to zero. Prove that if a set S ={v₁, v₂, ... , v_n} is a basis for V, then set S₁= {cv₁, cv₂, ... , cv_n} is also a basis of V.

Homework Equations

A set is a basis if it spans a subspace and is not collinear.

The Attempt at a Solution

S₁= {cv₁, cv₂, ... , cv_n} = c * {v₁, v₂, ... , v_n}

Since scalar c modifies all vectors equally, the set S₁ is a basis for vector space V.

 

[Chantry]

You have to prove:

a) It's linearly independent:
If S spans, that means ...
None of the vectors in S are a linear combination of another.
How can you prove this?
Think ..
av1 + bv2 + cv3 .. + Lvn = 0, implies a,b,c .. L = 0.
b): It spans.
If S spans V, that means that any vector of V can be represented by a linear combination of S.
How can you prove that cS also follows this?

The proof for b is similar to a ..