Show that the following are not vector spaces: (a) The set of all vectors [x, y] in R^2 with x ≥ y, with the usual vector addition and scalar multiplication. ------------------------------------------------[a b] (b) The set of all 2×2 matrices of the form [c d] in where ad = 0, with the usual matrix addition and scalar multiplication. I need help with this question. Could you please show your work and the solution.
Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of "addition" by (u, v) @ (x, y) = (u + x +1, v + y + 1) for all (u, v) and (x, y) in V. Define an operation of "scalar multipli- cation" by a® (x, y) = (ax, ay) for all a E R and (x,y) E V Under the two operations the set V is not a vector space....
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Let V, W and X be vector spaces. Let T: V -> W and S : W -> X be isomorphisms. Prove that SoT : V -+ X is an isomorphism.
Let V, W and X be vector spaces. Let T: V -> W and S : W -> X be isomorphisms. Prove that SoT : V -+ X is an isomorphism.
Question 1: Vector Spaces and Subspaces (a) Show that (C(0, 1]), R, +,), the set of continuous functions from [0, 1 to R equipped with the usual function addition and scalar multiplication, is a vector space. (b) Let (V, K, +,-) be a vector space. Show that a non-empty subset W C V which is closed under and - necessarily contains the zero vector. (c) Is the set {(x,y)T: z,y E R, y a subspace of R2? Justify.
4. Let A, X, Y, Z be normed vector spaces and B :X Y + Z be a bilinear map and f: A+X,9: A + Y be mappings that are differentiable at to E A. Show that the mapping 0 : A → Z, X HB(f(x), g(x)) is differentiable at Do and that dº(30)[h] = B(df (o)[N), 9(30))+ (f(x0), dg(xo)[h]) (he A).
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.
The components of a vector V ⃗ can be written ( V x , V y , V z ) . a) What are the components of a vector which is the sum of the two vectors, V ⃗ 1 and V ⃗ 2 , whose components are (8.5,−4.2,0.0) and (3.8,−7.5,−5.0) ? Enter the x, y, and z components of the vector separated by commas. b) What is the length of this vector? Express your answer using three significant figures.
1. Given a vector V with x- and y- components Vx = -6.2 and Vy = 2.9. What is the magnitude of vector V? 2. Given a vector V with x- and y- components Vx = -4.5 and Vy = 3. What is the direction of vector V with respect to the +x-axis counterclockwise?
(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...
Problem 3 (Inner Products). (a) Let V, W be two finite dimensional vector spaces, dim V = n, dim W-m and V x W-+ R be a bilinear function, i.e., for each a V and b E W: 1(a, r-Ay)-I(a,r) + λ|(a, y), for all r, y W, λ ε R and 1(u + λν, b)-1(u, b) + λ|(u, b), for all u, u ε ν, λ ε R. Thus for each fixed a E V, W 14-R is a...