9) Consider a vector space W where dim W= m (m-dimensional) Assume two subspaces V1, V2...
a. Let W and X both be subspaces of a vector space V. Prove that dim(WnX) > dim(W) + dim(X) - dim(V) b. Define a plane in R" (as a vector space) to be any subspace of dimension 2, and a line to be any subspace of dimension 1. Show that the intersection of any two planes in R' contains a line. c. Must the intersection of two planes in R* contain a line?
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).
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...
Q4 (5 points). Let V be a vector space with two subspaces S and T. Assume that dim(S) = dim(T) = 2 and SnT (O). Show that dim(V) 24
(4) Let {V1, V2, ..., Vn} be a basis for a vector space V. If w is an element of V whose coefficient vector is the zero vector, show that w must be the zero element.
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.
Let H = Span{V1, V2} and K = Span{V3,V4}, where V1, V2, V3, and V4 are given below. 1 V1 V2 V4 - 10 7 9 3 -6 Then Hand K are subspaces of R3. In fact, H and K are planes in R3 through the origin, and they intersect in a line through 0. Find a nonzero vector w that generates that line. W= [Hint: w can be written as C1 V2 + c2V2 and also as c3 V3...
(e) Let V and W be finite dimensional vector spaces, dim V-n and din W-m. Show that there is a bijection between the set Bil V, W;R) of all bilinear functions on V and W and the set GLn(m)R of all matrices of order n x m Hint: Uselal & rbl (e) Let V and W be finite dimensional vector spaces, dim V-n and din W-m. Show that there is a bijection between the set Bil V, W;R) of all...
Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}. (a) Consider the set S = {T : V -> F | T is a linear transformation}. Define the operations: (T1 + T2)(v) := T1(v) + T2(v), (aT1)(v) = a(T1(v)) for any v in V, a in F. Prove tat S with these operations is a vector space over F. (b) In S, we have elements fi : V -> F...
e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m. e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m.