Proving Vector Addition with Linear Independence

[b2386]

Hi all,

While working on my differential equations homework, I encountered a proof dealing with linear independence and vector addition. I sort of know how to proceed but, not having dealt with formal proofs much, I am afraid that I may not be addressing all necessary apects of the proof. Anyway, here is the question: Prove that if the vectors x = (x_1)i + (x_2)j and y = (y_1)i + (y_2)j
are linearly independent, then any vector z = (z_1)i + (z_2)j can be expressed as a linear combination of x and y.

The linear combination of x and y gives us (x_1)i + (x_2)j + (y_1)i + (y_2)j. Rearranging terms, [(x_1)+(y_1)]i + [(x_2)+(y_2)]j = x+y. We can now define x+y = z. Therefore, z = (z_1)i + (z_2)j

Where do I bring in the necessity of linear independence?

 

[radou]

Prove that if the vectors x = (x_1)i + (x_2)j and y = (y_1)i + (y_2)j are linearly independent, then any vector z = (z_1)i + (z_2)j can be expressed as a linear combination of x and y.

Actually, this question is kind of 'definition-like'. You have two vectors in V^2(O). Any set of two vectors in V^2(O) which are linearly independent (i.e. non collinear) form a basis for V^2(O), and hence every vector from V^2(O) can be represented uniquely as a linear combination of these two independent vectors.

 

[b2386]

What exactly is V^2(O)? I haven't had linear algebra so I am probably unfamiliar with some terminology.

EDIT: Is that just a 2-D vector space?

 

[radou]

What exactly is V^2(O)? I haven't had linear algebra so I am probably unfamiliar with some terminology.

My apologies for not pointing it out - V^2(O) (or call it whatever you like) is the set of all radius vectors in the Euclidean plane, where your story is, of course, set up in a Cartesian coordinate system.

EDIT: it could be, but be careful when using that terminology; formally, a 2-D vector space is any 2-dimensional vector space - its elements don't need to be radius vectors!