Question

Let T : V → V be a linear operator satisfying T2 = T. Define U1...

Let T : V → V be a linear operator satisfying T2 = T. Define U1 = {v ∈ V : T(v) = v} and U2 = {v ∈ V : T(v) = 0}. Prove that V = U1 ⊕ U2.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let T: V -> V be a linear map such that T2 - I = 0...
Let T: V -> V be a linear map such that T2 - I = 0 where I is the identity map on V. a) Prove that Im(T-I) is a subset of Ker(T+I) and Im(T+I) is a subset of Ker(T-I). b) Prove that V is the direct sum of Ker(T-I) and Ker(T+I). c) Suppose that V is finite dimensional. True or false there exists a basis B of V such that [T]B is a diagonal matrix. Justify your answer.
5. Let U1, U2, U3 be subspaces of a vector space V. Prove that U1, U2,...
5. Let U1, U2, U3 be subspaces of a vector space V. Prove that U1, U2, U3 are direct-summable if and only if (i) the intersection of U1 and U2 is 0.\, and (ii) the intersection of U1+U2 and U3 is 0. A detailed explanation would be greatly appreciated :)
Let U1, U2 be subspaces of a vector space V. Prove that the union of U1...
Let U1, U2 be subspaces of a vector space V. Prove that the union of U1 and U2 is a subspace if and only if either U1 is a subset of U2 or U2 is a subset of U1.
Let T be a linear operator such that T=D+N is a jordan decomposition of T. That...
Let T be a linear operator such that T=D+N is a jordan decomposition of T. That is, D is diagonalizable and N is Nilpotent. Show if S is an endomorphism of V, that is, if S is a linear operator with domain and codmain V, prove if T commutes with S, (ST=TS), then D commutes with S (DS=SD), and N commutes with S (NS=SN).
Let V be a vector space and let U1, U2 be two subspaces of V ....
Let V be a vector space and let U1, U2 be two subspaces of V . Show that U1 ∩ U2 is a subspace of V . By giving an example, show that U1 ∪ U2 is in general not a subspace of V .
Determine if the vector v is a linear combination of the vectors u1, u2, u3. If...
Determine if the vector v is a linear combination of the vectors u1, u2, u3. If yes, indicate at least one possible value for the weights. If not, explain why. v = 2 4 2 , u1 = 1 1 0 , u2 = 0 1 -1 , u3 = 1 2 -1
10 Linear Transformations. Let V = R2 and W = R3. Define T: V → W...
10 Linear Transformations. Let V = R2 and W = R3. Define T: V → W by T(x1, x2) = (x1 − x2, x1, x2). Find the matrix representation of T using the standard bases in both V and W 11 Let T :R3 →R3 be a linear transformation such that T(1, 0, 0) = (2, 4, −1), T(0, 1, 0) = (1, 3, −2), T(0, 0, 1) = (0, −2, 2). Compute T(−2, 4, −1).
Let B={u1,...un} be an orthonormal basis for inner product space V and v b any vector...
Let B={u1,...un} be an orthonormal basis for inner product space V and v b any vector in V. Prove that v =c1u1 + c2u2 +....+cnun where c1=<v,u1>, c2=<v,u2>,...,cn=<v,un>
Suppose V is a vector space and T is a linear operator. Prove by induction that...
Suppose V is a vector space and T is a linear operator. Prove by induction that for all natural numbers n, if c is an eigenvalue of T then c^n is an eigenvalue of T^n.
Let the linear transformation T: V--->W be such that T (u) = u2 If a, b...
Let the linear transformation T: V--->W be such that T (u) = u2 If a, b are Real. Find T (au + bv) , if u = (x, y) v = (z, w) and uv = (xz-yw, xw + yz) Let the linear transformation T: V---> W be such that T (u) = T (x, y) = (xy, 0) where u = (x, y), with 2, -3. Then, if u = ( 1.0) and v = (0.1). Find the value...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT