Direct sums, Polynomials

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August 31st 2008, 07:51 PM
mivanova

Direct sums, Polynomials

Hi,
Thank you so much for the help before. I appreciate it.
I have 2 new problems to solve. Please, help me!

1. Prove or disprove if U, V, W are subspaces of V for which
U (dir sum) W = V (dir sum) W then U=V

2. Suppose that p_0,...p_3 are polynomials in the space P_m(F) (of polynomials over F with degree at most m) such that p_j(2)=0 for eachj. Prove that the set {p_0,...,p_m} is not lin. independant in P_m(F).

Thank you in advance!
August 31st 2008, 08:56 PM
NonCommAlg

Quote:

Originally Posted by mivanova

1. Prove or disprove if U, V, W are subspaces of V for which
U (dir sum) W = V (dir sum) W then U=V[/tex]

ok, there's a problem with the names you gave your subspaces: you already assumed V is the vector sapce. so the subspaces should be named, say, U, W, and Z.

as a vector space over $\mathbb{R}$ let $V=\mathbb{R}^2$ and $e_1=(1,0), \ e_2=(0,1), \ e_3=e_1+e_2=(1,1).$ let $U=\mathbb{R}e_1, \ W=\mathbb{R}e_2, \ Z=\mathbb{R}e_3.$ then $V=U \oplus W = Z \oplus W.$ but $U \neq Z.$

Quote:

2. Suppose that p_0,...p_m are polynomials in the space P_m(F) (of polynomials over F with degree at most m) such that p_j(2)=0 for eachj. Prove that the set {p_0,...,p_m} is not lin. independant in P_m(F).

let $p_j(x) \in P_m(\mathbb{F}), \ j=0, \cdots, m,$ with a common root $\alpha.$ (in your problem $\alpha=2.$ ) then $q_j(x)=\frac{p_j(x)}{x-\alpha} \in P_{m-1}(\mathbb{F}), \ j=0, \cdots , m.$ but $\dim_{\mathbb{F}} P_{m-1}(\mathbb{F})=m,$ and

the set $\{q_j(x): \ j=0, \cdots, m \}$ has m + 1 elements. so it's linearly dependent, i.e. there exist $c_j \in \mathbb{F}, \ j=0, \cdots, m,$ not all zero such that $\sum_{j=0}^m c_jq_j(x)=0. \ \ \ \ (1)$

now multiply both sides of (1) by $x - \alpha$ to get $\sum_{j=0}^m c_jp_j(x)=0.$ thus $p_j(x), \ j=0, \cdots , m,$ are linearly dependent. Q.E.D.
September 1st 2008, 09:41 AM
mivanova

Direct Sums 2

Thank you so much for the help.
How would you prove the 1st problem
(1. Prove or disprove if U, V, W are subspaces of V for which
U (dir sum) W = V (dir sum) W then U=V[/tex])
in the general case (not only in R^2) but for any subspace of a vector sace?
Thanks!
September 1st 2008, 05:01 PM
NonCommAlg

Quote:

Originally Posted by mivanova

Thank you so much for the help.
How would you prove the 1st problem
(1. Prove or disprove if U, V, W are subspaces of V for which
U (dir sum) W = V (dir sum) W then U=V[/tex])
in the general case (not only in R^2) but for any subspace of a vector sace?
Thanks!

i did not prove it! i disproved it by giving a counter-example.