Linear Combination

Question

We define the 2x2 matrix

A = ( 3 5 )

2 2

If we multiply A and a vector u, we get a vector Au which can be seen as a linear combination of 2 vectors : except for the canonical basis, what obvious set of 2 vectors can it be?

Answer 1

Hello Kunal,

So I think that you're question is awesome. It is perfect for understanding 2 things. Firstly, how bases work, and secondly, how linear maps (a.k.a. matrices) behave.

To plainly answer your question, Au can be represented as a linear combination of any 2 vectors which form a basis. More precisely, if {v_1, v₂ } is any basis for R², then there exists scalars a₁ and a₂ such that

Au = a₁v₁ + a₂v₂.

What is an "obvious" linear combination? Given the context of the problem, you probably ought to use some property of the matrix A. What I suggest before going any further is actually multiplying it out. Still not clear? Think about using the linearity properties of A. I will discuss this latter method. Represent u in the standard basis.

That is,

u = u₁ e₁ + u₂e₂

where e₁ = (1, 0) and e₂ = (0, 1). Then using linearity, we have that

Au = A(u₁ e₁ + u₂e₂) = u₁ Ae₁ + u₂Ae₂ = u₁ a₁ + u₂a₂

where a₁ and a₂ are the first and second columns of the matrix A. Notice that we are using the fact that A is linear -- that we can pull out the scalars u₁ and u₂ and we can distribute the matrix A over addition.

So the answer to your question is that Au can be "obviously" represented as a linear combination of its columns. In your class, you will soon cover something called the "column space" of A, which is the set of all the vectors v which are "hit" by A -- that is, that there is some u such that Au = v. The column space of A happens to be the span of the columns of A.

Hope this helps!