- #1

Hall

- 351

- 88

- Homework Statement
- Main body is to be referred.

- Relevant Equations
- Main body should be referred.

##f : [0,2] \to R##. ##f## is continuous and is defined as follows:

$$

f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$

$$

f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$

##V = \text{space of all such f}##

What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V## (or those f’s can be constructed) is ##\{x, x^2\}##, and for ##x\in [1,2]## the basis for ##V## is ##\{x^3, x^2, x, 1\}##. I wonder how to combine it, and can I find the dimension of V without finding the basis for V?

$$

f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$

$$

f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$

##V = \text{space of all such f}##

What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V## (or those f’s can be constructed) is ##\{x, x^2\}##, and for ##x\in [1,2]## the basis for ##V## is ##\{x^3, x^2, x, 1\}##. I wonder how to combine it, and can I find the dimension of V without finding the basis for V?