Find a basis for this vector space

[Steve Turchin]

Homework Statement

Find a basis for the following vector space:

$V = \{ p \in \mathbb C_{\leq4} ^{[z]} | \ p(1)=p(i)$ and $p(2)=0 \}$
(Where $\mathbb C_{\leq4} ^{[z]}$ denotes the polynomials of degree at most 4)

Homework Equations

N/A

The Attempt at a Solution

I tried to find bi-terms with all possible degree combinations. such as:
$8z-z^4$ and $2z^3-z^4$
The $p(2)=0$ part is easy, but I can't seem to find any bi-terms that pass $p(1)=p(i)$
I'm afraid that randomly trying out tri-term and quad-term combinations can get messy.

And a side question: Is it true that, suppose there are no polynomials for which $p(1)=p(i)$, or more generally, a vector space that is the trivial one which contains only the zero vector. Then the basis of that vector space is the empty set?
Could you please help me? Thanks :)

 

[Mark44]

Homework Statement

Find a basis for the following vector space:

$V = \{ p \in \mathbb C_{\leq4} ^{[z]} | \ p(1)=p(i)$ and $p(2)=0 \}$
(Where $\mathbb C_{\leq4} ^{[z]}$ denotes the polynomials of degree at most 4)

Homework Equations

N/A

The Attempt at a Solution

I tried to find bi-terms with all possible degree combinations. such as:
$8z-z^4$ and $2z^3-z^4$
The $p(2)=0$ part is easy, but I can't seem to find any bi-terms that pass $p(1)=p(i)$
I'm afraid that randomly trying out tri-term and quad-term combinations can get messy.

And a side question: Is it true that, suppose there are no polynomials for which $p(1)=p(i)$, or more generally, a vector space that is the trivial one which contains only the zero vector. Then the basis of that vector space is the empty set?
Could you please help me? Thanks :)

Your polynomial has the form $p(z) = a_4z^4 + a_3z^3 + a_2z^2 + a_1z + a_0$. Since p(2) = 0, then z - 2 must be a factor, leaving the other factor a cubic.
So $p(z) = (z - 2)(c_3z^3 + c_2z^2 + c_1z + c_0)$
Use the given conditions that p(1) = p(i) to see if you can solve for the coefficients. Having one equation with four constants might be a clue for the size of the basis.

 

[Steve Turchin]

Thank you so much for the thorough reply and sorry for posting this on the wrong forum.
Solving $p(1)=p(i)$
$(1-2)(c_3+c_2+c_1+c_0)=(i-2)(c_3 i^3+c_2 i^2+c_1 i+c_0)$
I get: $-(c_3+c_2+c_1+c_0)+i(c_0-c_2+2c_3-2c_1)+c_3-c_1+2c_2-2c_0$
$c_0-c_2+2c_3-2c_1=0$

$2c_3+3c_2-c_0=0$
$2c_3+c_2-c_1=0$
Arbitrary values: $c_3=1,c_2=1,c_1=3,c_0=5$
So there's one polynomial: $p_1(z)= (z-2)(z^3+z^2+3z+5)$

Please correct me if I'm wrong: If there are four constants then I need four linear independent polynomial vectors. (or is it four linear independent vectors of constants?)

Here's another three:
$p_2(z)= (z-2)(2z^3+4z^2+8z+16)$
$p_3(z)= (z-2)(5z^3-9z^2+z-17)$
$p_4(z)= (z-2)(3z^3+z^2-7z+9)$

Got this far. How do I know if these polynomials are linearly independent, and $\mathsf {span}(p_1(z),p_2(z),p_3(z),p_4(z)) = V$

To check for linear independence, do I need to reduce a matrix, of the vectors of constants, to row echelon form, and make sure that there is only a single answer? If true, should I use the $c_3,c_2,c_1,c_0$ vectors or the $a_4,a_3,a_2,a_1,a_0$ vectors?
Is it true that if $\mathsf {dim}(p_1(z),p_2(z),p_3(z),p_4(z)) = 4$ and the the vectors are linearly independent, then that is the basis? By size of the basis, did you mean the dimension, that is equal to the amount of constants, in this case?

 

[vela]

Thank you so much for the thorough reply and sorry for posting this on the wrong forum.
Solving $p(1)=p(i)$
$(1-2)(c_3+c_2+c_1+c_0)=(i-2)(c_3 i^3+c_2 i^2+c_1 i+c_0)$
I get: $-(c_3+c_2+c_1+c_0)+i(c_0-c_2+2c_3-2c_1)+c_3-c_1+2c_2-2c_0$
$c_0-c_2+2c_3-2c_1=0$
$2c_3+3c_2-c_0=0$
$2c_3+c_2-c_1=0$

I can guess where you got $c_0-c_2+2c_3-2c_1=0$, but the origin of the following two equations is unclear. In any case, you can't claim $c_0-c_2+2c_3-2c_1=0$ because the coefficients are complex.

Like Mark, I initially thought to factor (z-2) out of the polynomial, but I think the problem is actually easier to do if you stick with the original version of f(z) in terms of the $a$'s. Write down the two equations you get from $f(2)=0$ and $f(1)-f(i)=0$. You'll have a system of equations, so you can solve for two coefficients in terms of the others. Perhaps once you see that, it'll jog your memory on how to find a basis.

 

[Steve Turchin]

You'll have a system of equations, so you can solve for two coefficients in terms of the others. Perhaps once you see that, it'll jog your memory on how to find a basis.

Thanks a lot vela!

$p(2)=16a_4+8a_3+4a_2+2a_1+a_0=0$

$p(1)-p(i)=0=a_4+a_3+a_2+a_1+a_0-(a_4-ia_3-a_2+ia_1+a_0)$

$(1+i)a_3+2a_2+(1-i)a_1=0 \ \ \Rightarrow \ a_3=\frac{(i-1)a_1-2a_2}{1+i}$

Solving the first equation:
$16a_4+8(\frac{i-1}{1+i}a_1-\frac{2}{1+i}a_2)+4a_2+2a_1+a_0 = 0$

Arbitrary values for $a_3,a_2,a_1:$
$a_1=1, \ a_2=-1, \ a_3=1$

$16a_4+8\cdot1+4\cdot(-1)+2\cdot1+a_0=0$
$16a_4+6+a_0 = 0$

Arbitrary values for $a_4,a_0:$
$a_4=-1 , a_0=0$

I get: $-z^4,z^3,-z^2,z,10$
The coefficients are real (subspace of complex). Do the coefficients have to be NOT real?
Is this a basis, or do I need several polynomials?

edit: Just noticed, the arbitrary values I chose are not linearly independent...edit 2 : Did some more calculations and came up with these three vectors:
$v_1= z^4-16$
$v_2= 4-8i+z^2-\frac{2}{1+i}z^3$
$v_3=-6+z-z^2+z^3$
Since in the system of coefficients, two coefficients can be solved in terms of the others.
I believe the basis should consist of three vectors.

 

[pasmith]

Homework Statement

Find a basis for the following vector space:

$V = \{ p \in \mathbb C_{\leq4} ^{[z]} | \ p(1)=p(i)$ and $p(2)=0 \}$
(Where $\mathbb C_{\leq4} ^{[z]}$ denotes the polynomials of degree at most 4)

Homework Equations

N/A

The Attempt at a Solution

I tried to find bi-terms with all possible degree combinations. such as:
$8z-z^4$ and $2z^3-z^4$
The $p(2)=0$ part is easy, but I can't seem to find any bi-terms that pass $p(1)=p(i)$

You need at least a quadratic, since if $p(1) = p(i)$ for linear $p$ then $p$ is constant, and in view of $p(2) = 0$ must be zero.

Therefore you can start with $$f(z) = a + (z - 1)(z - i)(Bz^2 + Cz + D)$$ as the most general polynomial of degree at most 4 which satisfies $f(1) = f(i)$.

But since you want $f(2) = 0$ it makes sense to write $$Bz^2 + Cz + D = \frac{b(z-2)^2 + c(z - 2) + d}{(2 - 1)(2 - i)} = \frac{b(z-2)^2 + c(z - 2) + d}{2 - i},$$ to obtain $$f(z) = a + \frac{(z - 1)(z - i)}{2 - i} \left( b(z - 2)^2 + c(z - 2) - a \right)$$ as the most general polynomial of degree at most 4 which satisfies $f(1) = f(i)$ and $f(2) = 0$.

 

[vela]

The coefficients are real (subspace of complex). Do the coefficients have to be NOT real?

No, they can be real.

edit 2 : Did some more calculations and came up with these three vectors:
$v_1= z^4-16$
$v_2= 4-8i+z^2-\frac{2}{1+i}z^3$
$v_3=-6+z-z^2+z^3$
Since in the system of coefficients, two coefficients can be solved in terms of the others.
I believe the basis should consist of three vectors.

Looks good. It sounds like you figured it out correctly.