Suppose V=U OU', where V is some vector space and U, U' CV are subspaces. Let...
QUESTION 8 Let V = U ㊥ W where V is a finite-dimensional vector space over a field F, and U and w are subspaces of V. Suppose U1 and U2 are subspaces of U and Wi and W2 are subspaces of W Show that QUESTION 8 Let V = U ㊥ W where V is a finite-dimensional vector space over a field F, and U and w are subspaces of V. Suppose U1 and U2 are subspaces of U...
Suppose that U and V are subspaces of a vector space W. Then UnV is a subspace of both U and V, and U and V are both subspaces of U +V. Show that (U+ V)/U ~ V/(UnV) Suppose that U and V are subspaces of a vector space W. Then UnV is a subspace of both U and V, and U and V are both subspaces of U +V. Show that (U+ V)/U ~ V/(UnV)
1. (10 points) Suppose that U and W are subspaces of a vector space V such that vi,, , ,tk İs a basis of U and wi,. . . , wn, V1, . , Uk is a basis of W. m, W1,.. ., Wn,v],.. . ,vk is a basis of U +W, and deduce that dim(U+W)- Show that u1,. .. , w1, dim(U) + dim(W) - dim(Unw).
Let W1 and W, be the subspaces of a vector space V. Show that WinW, is a subspace of V.
Let V be a vector space over a field F, and let U and W be finite dimensional subspaces of V. Consider the four subspaces X1 = U, X2 = W, X3 = U+W, X4 = UnW. Determine if dim X; <dim X, or dim X, dim X, or neither, must hold for every choice of i, j = 1,2,3,4. Prove your answers.
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)...
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...
Suppose I is a set, and let {Ui}iel be a collection of subspaces of the vector space V. Show that NU; CV LEI is also a subspace. Use V quantifiers in your answer.
8. Let UW CV be subspaces such that U eW=V. Show that Wis isomorphic to V/U. (Hint: Consider the projection map 1: V V/U restricted to W.)
6. Let E and F be subspaces of a vector space V. Prove that: (a) EUF V if and only if E V or F = V. _ (b) EUF is a subspace if and only if E C F or FCE