Question

Urgent!

Write down the definition of what it means for one vector to be a linear combination of a collection of other vectors.

Can a given vector v be written as a linear combination of vectors v1, v2, ...., vn in more than one way? Justify your answer.

This is Linear Algebra

Answer 1

In general, a linear combination is a particular way of combining things (variables, vectors, etc) using scalar multiplication and addition.

Working with vectors

Now back to vectors. Let’s say we have the following vectors:

v1=(123⎤⎦⎥v→1=[123], v⃗ 2=⎡⎣⎢351⎤⎦⎥v→2=[351], v⃗ 3=⎡⎣⎢008⎤⎦⎥v→3=[008]

What would linear combinations of these vectors look like? Well, a linear combination of these vectors would be any combination of them using addition and scalar multiplication.

Let the vectors v1,v2,v3,⋯vn be vectors in Rn and c1,c2,⋯,cn be scalars. Then the vector b, where b=c1v1+c2v2+⋯+cnvn is called a linear combination of v1,v2,v3,…vn. The scalars c1,c2,…,cn are commonly called the “weights”.

v can be written as a linear combination of v1, v2, v3.....vn in one and only one way.