Subspace notation help

1. Oct 24, 2004

jbmap

Hi everyone, I was hoping someone could help me with something. Could someone explain to me exactly what this expression means:

H = {(a+b) + (a - 2b)t + bt^2 | a E R, b E R}

the purpose is to find the dimension of the given subspace, which I know how to do, I have just never seen this notation, so I'm not exactly sure what this expresssion is telling me about the subspace. since a and e are real numbers, does that mean t is not necessarily a real number? and is the first part of the expression a single vector? or is it some sort of conglomeration of multiple vectors? Thanks for your help
Davy

2. Oct 24, 2004

Galileo

I think you are looking at a vector space of polynomials (of degree 2 or higher).
A set basis vectors could be: $\{1,t,t^2,t^3,...,t^n\}$
Then $$H=\{(a+b)+(a-2b)t +bt^2|a \in \mathbb{R}, b \in \mathbb{R}\}$$
would be a subspace of the polynomial space.
So, formally speaking, $t$ is not a number, but a vector.

3. Oct 24, 2004

HallsofIvy

Staff Emeritus
You also posted this under "linear algebra" and were given very good answers there.

The "notation" means that H is the subset of all quadratic polynomials
&alpha;+ &beta;t+ &gamma;t2 such that &alpha;= a-b, &beta;= a- 2b, and &gamma;= b.

The set of all quadratic polynomials has dimension 3 specifically because we can select any of the coefficients &alpha;, &beta;, &gamma; arbitrarily: a basis is { 1, t, t2}.

Here the coefficients depend upon only two numbers. Pick a or b arbitrarily and then you can calculate &alpha;, &beta;, &gamma; .

In particular, if you take a= 1, b= 0, the polynomial is 1+ t and if you take a= 0, b= 1, the polynomial is 1- 2t+ t2. A basis for H is {1+t, 1- 2t+ t2} so H has dimension 2.