1)Let W C R³ be the subspace generatedby the vectors (1,0, 1)and(1, 1,0).a )Find the orthogonal complement W+ with respect tousual or canonical inner product of R³. Therefore, itpresents a basis for W+.b)Make the same as the previous item considering, in place ofthe canonical internal product, the internal product:((x, y, 2), (x', y', z')) = 2xx' + yy' + z2'.

Given W⊂R3 be the subspace generated by the vectors (1,0,1) and (1,1,0). Solution a: Since every…

Answered: 1)Let W CR³ be the subspace generated…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4) Let V = M2×2 (C) with standard basis B = (m₁, m2, m3, m4) and let T : V → V be the map d - ¹ (( & d)) = ( 8 = & d) b + c Write down a basis for ^²(V), ^³(V) and ^^(V) Find the matrix for ^³(T) : ^³(V) → ^³(V) (hint: it will be a 4×4 matrix) Find the determinant of T (a) (b) (c)
The inner product in V is defined according to the formula (u, v) 1V2 - U₂v₁ + 2U₂V₂. Let u (4,1) and v = (2,3). i. Show that u and v form an orthogonal basis in the inner product space Use this basis to find an orthonormal basis by normalizing each vector. 11. Use the inner product defined in part b) to express the vector (1,1). as a linear combination of the orthonormal basis vectors obtained in part i. W: =
Let A be a 9x6 matrix with nullify equal to 1 a) find the rank and nullify of A^T b) suppose that the third column vector of A, col3 is equal to 0 determine the volume (among col1, col2, ....... col6) that form a basis for the range of A
Show that v, = (1,0,1) , v, = (1,1, 0) , v3 = (1,1,1) span R'. Express vector v= (2, 1, 3) as linear combination of v,,v,, V . 2
Let W be the subspace of IR' spanned by the vectors 1 4 -2 and -1 -7 Find the matrix A of the orthogonal projection onto W. A
In each part, find a basis for the given subspace of R', and state its dimension. a) All vectors of the form (a, b, 0, d). b) All vectors of the form (a, b, c, d) where a = b+d and c= 26 – d. c) All vectors of the form (a, b, c, d) where a = 26 = -c = 3d.
Suppose that V is a that has subspaces of U and W. Furthermore, suppose that {u¹, u2} is a basis for U, that {1, 2} is a basis for W and that the only vector that U and W have in common is the zero vector 0. Show that {u¹, u², w¹, w²}. After you get done, note where you used the fact that the only vector common to both U and W is 0. (Without this condition the vectors {u¹, u², w², w2} don't need to be linearly independent, so if you didn't use this condition you definitely made a mistake somewhere.)
The set of vectors {(1, 1, 0), (1, 0, 1), (0, 0, 1)} is an orthonormal basis for the space R^3 in some scalar product <.,.>. Then, the scalar product <(1, 1, 1), (1, 0, 0)> is equal to: (a) -1. (b) 0. (c) 1. (d) 2. (e) 4. The correct answer is (a). Why?
I know the basis converted to orthogonal is {<2, 1, -1, 1>,<11, -12, 5, -5>,<-1, -3, 0, 5>} and that the magnitude of these will be sqrt(7), sqrt(315), sqrt(35) - but I'm not sure how to use these to convert the basis to orthonormal.
a) Find a spanning set for the plane in R' given by x- 3y+4z=0 b) Find the matrix of the projection operator onto this plane. c) Find the projection of the vector v= (1,1,1) onto the plane. d) Find the projection of the vector w= (1,-1,-1) onto the plane.
Subspace W of R* is spanned by the vectors v1 = [1 – 2 3 – 1] and v2 = [1 1 – 2 3]1. - Then, what can be said about the subspace W? a) dim W = 1 b) dim W 2 c) dim W = 3 d) dim W = 4

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