Q7 8 Points Let V, W, and U be three finite dimensional vector spaces over R...
Q7 8 Points Let V, W, and U be three finite dimensional vector spaces over R and T:V + W and S : W + U be two linear transformations. Q71 4 Points Show that null(S o T) < null(T) + null(S) Please select file(s) Select file(s) Save Answer Q7.2 4 Points Show that rank(SoT) > rank(T) + rank(S) - dim(W) (Hint: Use part (1) at some point) Please select file(s) Select file(s) Save Answer
Let V, W, and U be three finite dimensional vector spaces over R and T:V + Wand S : W → U be two linear transformations. Show that rank( ST) > rank(T) + rank(S) - dim(W)
Let V, W, and U be three finite dimensional vector spaces over R and T:V + Wand S : W → U be two linear transformations. Show that null(SoT) < null(T) + null(S)
Q10.2 3 Points Let V and W be finite dimensional vector spaces over R and T:V + W be linear. Let Vo be a subspace of V and Wo = T(V). (Select ALL that are TRUE) If T is surjective then Vo = {v E V: there is w E Wo such that T(v) = w}. If T is injective then dim(V.) = dim(Wo). dim(ker(T) n ) = dim(V.) - dim(Wo). Save Answer
Let V and W be finite dimensional vector spaces over R and T:V + W be linear. Let V be a subspace of V and Wo = T(V). (Select ALL that are TRUE) If T is surjective then Vo = {v EV : there is w E Wo such that T(v) = w} If T is injective then dim(VO) = dim(W). dim(ker(T) n Vo) = dim(VO) - dim(Wo).
Q9 11 Points Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S: W → V a generalized inverse of Tif To SoT=T and SoTo S = S. Q9.3 4 Points If V and W are finite dimensional, show that there exists a generalized inverse of T. Please select file(s) Select file(s) Save Answer
Q9 11 Points Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S: W+V a generalized inverse of Tif To SoT = T and Soto S=S. 09.1 3 Points If T is an isomorphism, show that T-1 is the unique generalized inverse of T. Please select file(s) Select file(s) Save Answer Q9.2 4 Points If S is a generalized inverse of T show that V =...
Problem 6. Let V, W, and U be finite-dimensional vector spaces, and let T : V → W and S : W → U be linear transformations (a) Prove that if B-(Un . . . , v. . . . ,6) is a basis of V such that Bo-(Un .. . ,%) s a basis of ker(T) then (T(Fk+), , T(n)) is a basis of im(T) (b) Prove that if (w!, . . . ,u-, υ, . . . ,i)...
Problem 5. Let W and U be finite-dimensional vector spaces, and let T : W > W and S : W -> U be linear transformations. Prove that if rank(S o T) L W W such that S o T = So L. = rank(S), then there exists an isomorphism (,.. . , Vk) is a basis of ker(T), and let (w1, ., wr) is a basis of im(T) nker(S) if 1 ik Hint: Let B (vi,... , Vk,...,vj,) be...
Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S: W → V a generalized inverse of T if To SOT = T and SoTo S = S. If V and W are finite dimensional, show that there exists a generalized inverse of T.