Here
is the required answer.Hope this helps you.Please upvote my work if
you get benefited.Your feedback is very much precious.Thank
You.
14. (5 points) Let W be a subspace of V. Let Ow be the zero vector...
Q9. Let W be a subspace of R". (a) Prove that w+ is a subspace of R". (b) Prove that if a vector v belongs to both W and W+, then v must be the zero vector.
1
Problem 4. Let V be a vector space and let U and W be two subspaces of V. Let (1) Prove that ifU W andWgU then UUW is not a subspace of V (2) Give an example of V, U and W such that U W andWgU. Explicitly verify the implication of the statement in part1). (3) Proue that UUW is a subspace of V if and only if U-W or W- (4) Give an example that proues the...
please proof and explain. thank you 1. Let W be a finitely generated subspace of a vector space V . Prove that W has a basis. 2. Let W be a finitely generated subspace of a vector space V . Prove that all bases for W have the same cardinality.
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.
Problem 4. Let V be a vector space and let U and W be two subspaces of V. Let (1) Prove that if U g W and W g U then UUW is not a subspace of V 2) Give an ezample of V, U and W such that U W andW ZU. Explicitly verify the implication of the statement in part (1) (3) Prove that UUW is a subspace of V if and only ifUCW or W CU.' (4)...
Problem 3. Let V and W be vector spaces, let T : V -> W be a linear transformation, and suppose U is a subspace of W (a) Recall that the inverse image of U under T is the set T-1 U] := {VE V : T(v) E U). Prove that T-[U] is a subspace of V (b) Show that U nim(T) is a subspace of W, and then without using the Rank-Nullity Theorem, prove that dim(T-1[U]) = dim(Unin (T))...
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).
(5 points) Let 5 -4 v= 1-3 -3 and let W the subspace of R4 spanned by ū and 7. Find a basis of W?, the orthogonal complement of W in R4.
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y...
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...