Finding a basis

**chella182** · December 11th 2009, 02:24 AM

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

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 $f(0)=f(1)=f(-1)$ .

**HallsofIvy** · December 11th 2009, 04:55 AM

Originally Posted by chella182

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

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 $f(0)=f(1)=f(-1)$ .

Any "polynomial of degree at most 5" can be written as $f(x)= a+ bx+ cx^2+ dx^3+ ex^4+ fx^5$ . f(0)= f(1) means that $a= a+ b+ c+ d+ e+ f$ . f(0)= f(-1) means that $a= a- b+ c- d+ e- f$ .
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.

**Raoh** · December 11th 2009, 05:24 AM

Originally Posted by chella182

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

rewrite your set as,
$\left \{ a(1,0,1,0)+b(0,1,0,1)+c(1,0,0,1)+d(0,1,1,0),(a,b,c ,d)\in \mathbb{R}^{3} \right \}<br />$

**chella182** · December 11th 2009, 06:55 AM

Originally Posted by HallsofIvy

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?

Yeahh don't you think I tried that?

I just don't like pure maths and none of it made sense when I read over the notes / from books I find. I don't know how to find a linearly independent supspace, either. I just about understand what linearly independent is, so I think this module's a lost cause

Originally Posted by HallsofIvy

Any "polynomial of degree at most 5" can be written as $f(x)= a+ bx+ cx^2+ dx^3+ ex^4+ fx^5$ . f(0)= f(1) means that $a= a+ b+ c+ d+ e+ f$ . f(0)= f(-1) means that $a= a- b+ c- d+ e- f$ .
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.

Okay, I get that... but I don't understand why I'm doing any of this

like why am I setting one equal to 1 and the rest equal to 0? This is what I need explaining. Like I say, think I'm just gonna leave it until I can get proper help from someone next semester; I've been busting my arse over this work all week and gotten no where. Thanks, though...

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