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15. If ū + ✓ + W = , and W = -50j, find ū and ū. Show work. (15, ū =) u 45° 27° (15, ✓ =) O w 5
(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u, v e R". Show that ul if and only if ||ü + 해2 (c) Let W be a subspace of R" with an orthogonal basis {w1, ..., w,} and let {ö1, ..., ūg} 22 orthogonal basis for W- (i) Explain why{w1, ..., üp, T1, .., T,} is an (ii Explain why the set in (i) spans R". (iii Show that dim(W) + dim(W1) be...
Exercise 1. Let v = 2 ER3. Recall that the transposed vector u is ū written in row form, 3 that is, of = [1 2 3]. It can be seen as a 1 x 3 matrix. For every vector R3, set f(w) = 1 WER. (i) Show that f: R3 → R defines a linear transformation. (ii) Show that f(ū) > 0. (iii) What are the vectors we R3 such that f(w) = 0?
Given the following vectors: ū= 3 ū= W = > (a) Find the projection of ū onto ū. BOX YOUR ANSWER. (b) Find the projection matrix of the projection in part (a). BOX YOUR ANSWER. (c) Find the projection of ū onto the subspace V of R3 spanned by ✓ and W. (You may use MATLAB for matrix multiplication in this part, but you must provide the expressions in terms of matrices.) BOX YOUR ANSWER. (d) Find the distance from...
5 3 1 Let ū = < 2,-3> V = <-2,0 > w = <3,3 > Graph vectors ū, ū, and w in standard position with corresponding terminal points, A, B, and C, respectively. (72 point) What is the length of the altitude of AABC from vertex A? (72 point) -5 -3 -1 -1 0 1 3 5 -3 -5
1. Let {ü, 7,w, i}, where u = (3,-2), v = (0,4), ū = (-1,5) and i = (-6,4). Find the components of the resultants obtained by doing the following linear combinations. a. r = 2ū - 40 b. š= 3ū – +20 +
11. Solve this boundary value problem for u(x, t): n2 xu,-(x14),--11 (0<x <c,0 11 (c, 1) = 0, u(x, 0) = f(x), where u is continuous for0sxc,0 and where n is a positive integer. Answer: u(x, 1) Σ A,Jn(gjx) exp (-α,1), where a", and A, are the constants j-1
11. Solve this boundary value problem for u(x, t): n2 xu,-(x14),--11 (0
Solve the PDE using laplace: (dw/dx)+(x*(dw/x)) =xu(t-1) w(x,0)=0 if x>=0 and w(0,t)=0 if t<=0
+X Show het Xu Exp)
+X Show het Xu Exp)
Prove that each of the metrics (4.1) , (4.2) , and (4.3) has
property (4.4) .
211/2 4.2) (4.3) A sequence {xu) = (xe)-1 converges to x = (지,. . .x,) in X if and only if for each k the sequence of component entries {XkU)}-i converges to x, in Xk. (4.4)
211/2
4.2) (4.3)
A sequence {xu) = (xe)-1 converges to x = (지,. . .x,) in X if and only if for each k the sequence of...