Homework Answers
2.6.22.
a. TW : Rn→R is defined by TW (x) = w.x for all x in Rn.
Let x and y be 2 arbitrary vectors in Rn and let k be an arbitrary real scalar. Then TW (x+y) = w.(x+y) = w.x+w.y = TW(x)+ TW(y). Thus, TW preserves vector addition.
Also, TW(kx) = w.(kx) = k(w.x) = k TW(x). Thus, TW preserves scalar multiplication.
Hence , TW is a linear transformation.
(b). Let x = (x1,x1,…,xn) and let T Rn→R be a linear transformation defined by T (x) = a1 x1+a2 x2+…+an xn, where a1, a2,…,an are real scalars. Then T(x) = TW (x) where w= (a1, a2,…,an).
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Exercise 2.6.22 Given vectors w and x in R", denote their dot product by w x....
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Please answer this question by typing or writing clearly with
explanations
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Let V, W, and U be three finite dimensional vector spaces over R and T:V +...
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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?
Hi please give a full solution Let W denote the set {(x, y, z) e Rº...
Hi please give a full solution
Let W denote the set {(x, y, z) e Rº | xy >0} Which of the following statements are correct? (There can be several correct statements.) W is closed under multiplication by a scalar The zero-vector belongs to W W is closed under addition of vectors
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Let T: P2 --> R2 be the linear transformation such that T(x+1)=(1,1), T(x2)=(1,0) and T(x-1)=(0, 1). Find T(2+x+x2).
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Linear algebra please prove and write neatly Any set of m vectors in R™ is linearly...
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Please answer this question by typing or writing clearly with
explanations
3. Define T: M >R be the linear transformation defined by (a-b,c+2d) T d a. Find ker(T) b. Give TWO examples of vectors in the kernel. c. Is T one-to-one? Explain.
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)
Dot Product To use the dot product to find the components of forces acting in arbitrary directions. A three-legged structure is shown below. It is attached to the ground at points B (4.92, 0, 0) and (-4.92, 0, 0) and connected to the ceiling at point A = (-5.42 3.46 4.76) The three legs connect at point D- (0,2.62, 4.00), where two forces, F and P, act. Force F is given by F 5.00 i 12.0j +24.8 k N; P...
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?
Hi please give a full solution
Let W denote the set {(x, y, z) e Rº | xy >0} Which of the following statements are correct? (There can be several correct statements.) W is closed under multiplication by a scalar The zero-vector belongs to W W is closed under addition of vectors
Let T: P2 --> R2 be the linear transformation such that T(x+1)=(1,1), T(x2)=(1,0) and T(x-1)=(0, 1). Find T(2+x+x2).
Q4: The switch in the has been open a long time before closing at t=0. Find iz(t) for t> 0. t = 0 2.5 k 2 il 1362.5H 15V 2.5 UF 9 mA
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Linear algebra please prove and write neatly
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