Linear Algebra

[think.ms.tink]

Let (Beta1, Beta2, Beta3) be a basis for a vector.
Show that (c1Beta1,c2Beta2,c3Beta3) is a basis when c1,c2,c3 cannot equal zero. What happens when at least one ci is 0? Prove that (alpha1, alpha2, alpha3) is a basis where alphai=beta1+betai?

I am not sure how to put this together all i know is that if c1,c2,c3 equal zero, then the whole thing would be zero (0,0,0), correct? Thank you.

 

[pka]

In this space any linearly independent set of three vectors is a basis.
If \(\displaystyle \left\{ {c_1 \beta _1 ,c_2 \beta _2 ,c_3 \beta _3 } \right\}\) were a linearly dependent set then one of the three is a linear combination of the other two.
But that means the original \(\displaystyle \left\{ { \beta _1 , \beta _2 , \beta _3 } \right\}\) is also linearly dependent.
That contradicts the that it is a basis

 

[think.ms.tink]

so if ci =0 does that mean everything goes to the null space.

 

[daon]

If any single c_i is zero, it is a dependent set. No set containing the zero vector can form a basis. The 0 vector is a linear combination of any set of vectors.