Given that
is linear and
is linear . Now consider
then
since T is linear
Since S is linear
Now we consider
Since T is linear
Since S is linear
Hence is
linear
Are the Resulting Functions Linear? At this point you should wonder whether these combinations of functions...
please help me with questions 1,2,3
1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
6. Let S and T both be linear transformations from a vector space V to itself. Let W be the set {v€ V: S(v) = T(v) }. Prove that W is a subspace of V.
Please answer the following
question. Thank you.
30. Let T:V W be a linear transformation from a vector space V into a vector space W.Prove that the range of T is a subspace of W.[ Hint: Typical elements of the range have the form T(x) and T(w) for some x, w in V.]
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
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
only a-i T or F
lit khd where it came from 4. You do not need to simplify results, unless otherwise stated. 1. (20pts.) Indicate whether each of the following questions is True or False by writing the words "True" or "False" No explanation is needed. (a) If S is a set of linearly independent vectors in R" then the set S is an orthogonal set (b) If the vector x is orthogonal to every vector in a subspace W...
(1 point) Determine whether the given set S is a subspace of the vector space V. A. V = R", and S is the set of solutions to the homogeneous linear system Ax = 0 where A is a fixed mxn matrix. B. V is the vector space of all real-valued functions defined on the interval (-oo, oo), and S is the subset of V consisting of those functions satisfying f(0) 0 C. V Mn (R), and S is the...
). owevel,s Haider to lactor It. Sol am giving that det ( A-U)=(A-1)(A+2) (A+2) . Find the eigenvalues Find each eigenspace. Find a basis for each eigenspace. following problem is given so you have more practice in proving that a set i understand what it means to belong to X. lem 4: Let T be a linear transformation from a vector space V to a vector space W That is T: V > W. Let S be a subspace of...
Problem 13. For each of the following we are given two vectors u, we V and a linear trans- formation from a vector space V to itself. Check if the given vectors are eigenvectors for the transformation. If yes, then find the corresponding eigenvalues. (a). V=P3, 7(p(x))=x?p"(x) — xp'(x), with u =2+3x? and w=x?. (b). V = Muj, T(A)=A+A”, with u=[?) and w=[; ?] (c). V = P2, L(P(x) p(x)dx + (x – 3)p'(x) with u = 100 and w=3+3x.