1. ## Finding a basis

Find a basis for the space $\displaystyle \{(a+c, b+d, a+d, b+c)\vert a,b,c,d \epsilon \mathbb{R}\}$

Again, literally no clue on this; my lecturer really can't teach very well. Also, might as well put the other problem I have in here; it's in the same vein (I think!)

Find a basis for the space of all polynomials of degree at most 5 for which $\displaystyle f(0)=f(1)=f(-1)$.

2. Originally Posted by chella182
Find a basis for the space $\displaystyle \{(a+c, b+d, a+d, b+c)\vert a,b,c,d \epsilon \mathbb{R}\}$

Again, literally no clue on this; my lecturer really can't teach very well. Also, might as well put the other problem I have in here; it's in the same vein (I think!)
Then look up the definitions for yourself! What vector would you get if you take a= 1, b= c= d= 0? What about b= 1, a= c= d= 0? Also try c= 1, a= b= d= 0 and d= 1, a= b= c= 0. Are those vectors linearly independent? If not, can you find a linearly independent subspace?

Find a basis for the space of all polynomials of degree at most 5 for which $\displaystyle f(0)=f(1)=f(-1)$.
Any "polynomial of degree at most 5" can be written as $\displaystyle f(x)= a+ bx+ cx^2+ dx^3+ ex^4+ fx^5$. f(0)= f(1) means that $\displaystyle a= a+ b+ c+ d+ e+ f$. f(0)= f(-1) means that $\displaystyle a= a- b+ c- d+ e- f$.
With those two equations you should be able to write two of the coefficients in terms of the other four. Now choose one of those four to be 1, the others 0, etc.

3. Originally Posted by chella182
Find a basis for the space $\displaystyle \{(a+c, b+d, a+d, b+c)\vert a,b,c,d \epsilon \mathbb{R}\}$
$\displaystyle \left \{ a(1,0,1,0)+b(0,1,0,1)+c(1,0,0,1)+d(0,1,1,0),(a,b,c ,d)\in \mathbb{R}^{3} \right \}$
Any "polynomial of degree at most 5" can be written as $\displaystyle f(x)= a+ bx+ cx^2+ dx^3+ ex^4+ fx^5$. f(0)= f(1) means that $\displaystyle a= a+ b+ c+ d+ e+ f$. f(0)= f(-1) means that $\displaystyle a= a- b+ c- d+ e- f$.