9.4 (Idempotent endomorphisms) Let V be an n-dimensional vector space over Kand let End(V) such t...
• Let V be a 2-dimensional real vector space, and let T E End(V). Show that T is diagonalizable over C but not over R if and only if tr(T)2 < 4. det(T).
Q-) Let F be an object ond V is a finite dimensional vector Space on the object. . that if v is linear trons formation, ronkt is zero a) Show or 1. b) If Liv> v is linear tronsformation, show that ker L c ker L² and L(v) 2 L² (v). ( Note : L²=LoL and ker L, be defined as subspace of L.).
Question 1. Let V be a finite dimensional vector space over a field F and let W be a subspace of Prove that the quotient space V/W is finite dimensional and dimr(V/IV) = dimF(V) _ dimF(W). Hint l. Start with a basis A = {wi, . . . , w,n} for W and extend it to a basis B = {wi , . . . , wm, V1 , . . . , va) for V. Hint 2. Our goal...
Let W be a subspace of an n-dimensional vector space V over C, and let T:V V be a linear transformation. Prove that W is invariant under T if and only if W is invariant under T- I for any i EC.
3. Let Te L(V), where V is a finite-dimensional C-vector space. Prove that T is diago- nalizable if and only if Ker(T – a id) n Im(T - a id) = {0} for all a E C.
Let pEN be a prime and let V be a vector space over Zp with dimension n. Show that |v| = pn. Let pEN be a prime and let V be a vector space over Zp with dimension n. Show that |v| = pn.
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
linear algebra please help on both questions 2. Let V be an n-dimensional vector space over C. Classify, up to similarity, all JE C(V), where2-Idy 3. Recall in assignment 2, no. 7, you showed UoM-lv where U, M E C(V), V-Fİrl,Ms Mr maps p to ap, and U maps 1 to 0 and a to for nEN a. Show that 0 E σ(M). b. Show 0 is not an eigenvalue of M. c. Define an inner product on V: Flr]...
Problem #6. Let V be a finite dimensional vector space over a field F. Let W be a subspace of V. Define A(W) e Vw)Vw E W). Prove that A(W) is a subspace of (V).
Problem 4. Suppose V is an n-dimensional complex vector space and TEL(V) is such that dim ker(T"-k) #dim ker(T"-k+1) for some k <n-1. Show that I has at most k eigenvalues. Hint: Is zero an eigenvalue? What is its geometric multiplicity? Solution: Write your solution here.