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 U, W be subspaces of R" and assume UnW = {0„}. Prove that dim(U + W) = dim(U)+ dim(W). Here U + W = {u+w:u EU,w e 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
Consider the three vectors -(1)-~-(-) -2 u= 3 3 -(:). 9 0 3 W = -3 Let S denote the one dimensional subspace spanned by u, and let S denote its orthogonal complement. (a) Compute the matrix Proju which is the orthogonal projection onto S. Justify your answer using words. (b) Find the vector in S that is closest to v. Justify your answer using words. (c) Find the vector in S that is closest to w. Justify your answer using words.
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
V is a 3-dimensional vector space with basis B = (vı, V2, v3). Let w = vi + v2 + v3, w2 = v2+V3, W3 = v1 +v2+2v3 and set B' = (w1, w2, W3), which is also a basis of V. a) Determine the change of basis matrix from B' to B, Pg-→B. b) Determine the change of basis matrix from B to B', P3¬B'.
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.
Consider the Euclidean inner product space R³ with a basis B={(1,1,1),(0,1,1),(0,0,1)}. Find an orthonormal basis of R³. 0^{(1,2,1),(0,-1,-1),(0,1,0)) 08((1.1.1).(-2,1,1),(0,1,1)) 0(-2,1,1),(0,1,1),(,-1,1)} O A. O B. OD. None in the given list O E. {(1,1,1),(0,0,1),(0,-1,1)}
Find a basis for the subspace of R° that is spanned by the vectors V1 = (1, 0, 0), v2 = (1, 0, 1), v3 = (5, 0, 1), v4 = (0, 0, -2) V2 and v4 form a basis for span {v1, V2, V3, V4}. V1 and v4 form a basis for span {v1, V2, V3, V4}. V1 and v3 form a basis for span {v1, V2, V3, V4}. Ov1 and v2 form a basis for span {v1, V2, V3, V4}. V2 and v3 form a basis for span {v1, V2, V3, V4}. V3 and v4 form a basis for span {v1, V2, V3, V4}.
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.)

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