Let V⊂R^4 be the subspace defined by the equation x1 + 3x2 - 5x3 - x4 = 0. a) Find an orthogonal basis for V. b) Which is the point over the plane x1 + 3x2 - 5x3 - x4 = 36 closest to the origin?

Question

Question

Let V⊂R^4 be the subspace defined by the equation x1 + 3x2 - 5x3 - x4 = 0.

a) Find an orthogonal basis for V.

b) Which is the point over the plane x1 + 3x2 - 5x3 - x4 = 36 closest to the origin?

Answer 1

Answer #1

0 (5)00 0 (-5)0 orthogonal basis is,

by uisng lagranges multipliers 2x1 + λ = 0; 2x2 +32 0; 31 5λ xi +3x2-5x3-x,-36

Answer 2

Similar Homework Help Questions

6. Let P be the subspace in R 3 defined by the plane x − 2y...

6. Let P be the subspace in R 3 defined by the plane x − 2y + z = 0. (a) [5 points] Use the Gram–Schmidt process to find orthogonal vectors that form a basis for P. (b) [5 points] Find the projection p of b = (3, −6, 9) onto P. 6. Let P be the subspace in R3 defined by the plan 2y+z0 (a) [5 points] Use the Gram-Schmidt process to find orthogonal vectors that form a basis...
Find a basis of the subspace of ℝ3 R 3 defined by the equation 6x1−5x2−8x3=0 6...

Find a basis of the subspace of ℝ3 R 3 defined by the equation 6x1−5x2−8x3=0 6 x 1 − 5 x 2 − 8 x 3 = 0 . (1 point) Find a basis of the subspace of R3 defined by the equation 6x1 5x2 -8x3-0. Basis:
e, none of these 7. Let {1,..., up} be an orthogonal basis for a subspace W...

e, none of these 7. Let {1,..., up} be an orthogonal basis for a subspace W of R" and {...., } be an orthogonal basis for Wt. Determine which of the following is false. a. p+q=n b. {U1,..., Up, V1,...,0} is an orthogonal basis for R". c. the orthogonal projection of the u; onto W is 0. d. the orthogonal projection of the vi onto W is 0. e. none of these 8. Let {u},..., up} be an orthogonal basis...

Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the...

Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the first three orthogonal polynomials on -1,1. Let f(x) 21, and consider the subset G-{g(z) | 〈f,g〉 0), the set of functions orthogonal to f using the L inner product on, (This can be thought of as the plane normal to f(x) in the three-dimensional function space.) Let h(z) 2-1. Find the function g(x) є G in the plane which is closest to h(x)....
(1 point) Find the orthogonal projection of -1 -5 V = 9 -11 onto the subspace...

(1 point) Find the orthogonal projection of -1 -5 V = 9 -11 onto the subspace V of R4 spanned by -4 -2 -4. -5 X1 = and X2 1 -28 -4 0 projv(v)
2. Find the closest point to y = in the subspace H = Span [ o།...

2. Find the closest point to y = in the subspace H = Span [ o། [ 17 [10] 3. Let B = {| 2 |,|-2， 1}. Find the coordinate vector of x = [1] relative to the [=1] [4] [2] orthogonal basis B for R3. ངོ- v1cs None of the above 5. Which of the following is true about the sets of vectors S and T? 3 1 [3 ] , 2 ), T={l U L-13] The set S...

6 consider the subspace of R² defined by 12+ana+z=o. Let B=Cui, 3) be the basis of...

6 consider the subspace of R² defined by 12+ana+z=o. Let B=Cui, 3) be the basis of v, where [3] [:] If [x]-[-3], find 2. check all that apply. R-6-3 [:] or can not be found as it is not inv D 4 -3 B 2 » D { 4 -3.
Find the orthogonal projection of v⃗ 26 11 8 4 0 (1 point) Find the orthogonal...

Find the orthogonal projection of v⃗ 26 11 8 4 0 (1 point) Find the orthogonal projection ofv- 0 onto the subspace V of R spanned by and 28 (Note that these three vectors form an orthogonal set.) projv (u)-
4. Let v={[a -.:a,nccc} Note that V is a vector space over R. View V as...

4. Let v={[a -.:a,nccc} Note that V is a vector space over R. View V as a R-vector space. (a) Find a basis for V over R. (b) Let W be the set of all matrices M in V such that M21 = -M12, where denotes complex conjugate. Show that W is a subspace of V over R and find a basis for Wover

Problem 7: Let S be the subspace of R' defined by the equation: x,+2x2-13 = a)...

Problem 7: Let S be the subspace of R' defined by the equation: x,+2x2-13 = a) Find an orthonormal basis for S and an orthonormal basis for S b) Find the vectors liE S and vES® such that the vector x = (2,1,-8/ can be written in the form x = 11 +-