1. ## Vector Spaces

Which of the following are vector spaces?

(a) The set of all continuous functions on the interval $\displaystyle [0,1]$.

(b) The set of all non-negative functions on the interval $\displaystyle [0,1]$.

(c) The set of all polynomials of degree exactly $\displaystyle n$.

(d) The set of all symmetric $\displaystyle n \times n$ matrices, i.e. the set of matrices $\displaystyle A = \{a_{j,k} \}_{j,k = 1}^{n}$ such that $\displaystyle A^{T} = A$.

So (a) is not a vector space because there is no $\displaystyle \bold{0}$ vector?

(b) is a vector space.

(c) is not a vector space.

(d) is a vector space.

Are these correct?

2. Originally Posted by horacejerry
Which of the following are vector spaces?

(a) The set of all continuous functions on the interval $\displaystyle [0,1]$.

(b) The set of all non-negative functions on the interval $\displaystyle [0,1]$.

(c) The set of all polynomials of degree exactly $\displaystyle n$.

(d) The set of all symmetric $\displaystyle n \times n$ matrices, i.e. the set of matrices $\displaystyle A = \{a_{j,k} \}_{j,k = 1}^{n}$ such that $\displaystyle A^{T} = A$.

So (a) is not a vector space because there is no $\displaystyle \bold{0}$ vector?
so, f(x) = 0 is not a continuous function on [0,1]?

(b) is a vector space.
oh, and what would the additive inverse of say, f(x) = 1, be?

(c) is not a vector space.
what condition failed?

(d) is a vector space.
ok

3. (c) is not a vector space.
Originally Posted by Jhevon
what condition failed?
This vector space isn't closed under addition...

Ex: $\displaystyle \bold u=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0$ and $\displaystyle \bold v=-a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0$

Thus, $\displaystyle \bold u\oplus\bold v=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0$ $\displaystyle +\left(-a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0\right)$ $\displaystyle =2a_{n-1}x^{n-1}+...+2a_2x^2+2a_1x+2a_0\notin V$

--Chris

4. Originally Posted by Chris L T521
This vector space isn't closed under addition...

Ex: $\displaystyle \bold u=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0$ and $\displaystyle \bold v=-a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0$

Thus, $\displaystyle \bold u\oplus\bold v=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0$ $\displaystyle +\left(-a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0\right)$ $\displaystyle =2a_{n-1}x^{n-1}+...+2a_2x^2+2a_1x+2a_0\notin V$

--Chris
yes, that is one reason why.

i wanted to know if the poster knew that though