Let W be the set of all vectors of the form {(a,b,c):a-b=c). show that @ is a subspace of R^3
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2s+2t 3s . Show that W is a subspace of R Let W be the set of all vectors of the form by finding vectors u and v such that W = Span{u,v). 3s 4t Write the vectors in W as column vectors. EHRIE 2s +2t 3s #su + tv 3s 4t
Let w be a subspace of R", and let wt be the set of all vectors orthogonal to W. Show that wt is a subspace of R" using the following steps. a. Take z in wt, and let u represent any element of W. Then zu u = 0. Take any scalar c and show that cz is orthogonal to u. (Since u was an arbitrary element of W, this will show that cz is in wt.) b. Take z,...
Linear Algebra Advanced Let A be vectors in R". Show that the set of all vectors B in R" such that B is perpendicular to A is a subspace of R". In other words shovw W Be R"IA B-0 for a vector Ae R" is a subspace.
Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orthonormal set using the dot product on R. (Hence it forms an orthonormal basis for W.) (b) Let w = (-1,1,5,5) EW. Using the formula in the box above, express was a linear combination of u and u. (c) Let v = (-1,1,3,5) = R'. Find the orthogonal projection of v onto W.
Problem 1: consider the set of vectors in R^3 of the form: Material on basis and dimension Problem 1: Consider the set of vectors in R' of the form < a-2b,b-a,5b> Prove that this set is a subspace of R' by showing closure under addition and scalar multiplication Find a basis for the subspace. Is the vector w-8,5,15> in the subspace? If so, express w as a linear combination of the basis vectors for the subspace. Give the dimension of...
(3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A- (3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A-
(10) Let ū ER. Show that M = {ū= | ER*:ūū= 0) is a subspace of R'. Definition: (Modified from our book from page 204.) Let V be a subspace of R". Then the set of vectors (61, 72, ..., 5x} is a basis for V if the following two conditions hold. (a) span{61, 62,...,x} = V (b) {61, 62, ..., 5x} is linearly independent. Definition: Standard Basis for R" The the set of vectors {ēi, 72, ..., en) is...
3t Let W be the set of all vectors of the form 5 +5 5s Show that W is a subspace of R* by finding vectors u and v such that W=Span{u,v). 5s Write the vectors in Was column vectors 31 5 4 5t = su + tv 5s 5s What does this imply about W? O A. W = Span(u,v} OB. W = Span{s.t O C. Ws+t OD. W=u+v
(12 points) Let vi = 1 and let W be the subspace of R* spanned by V, and v. (a) Convert (V. 2) into an ohonormal basis of W NOTE: If your answer involves square roots, leave them unevaluated. Basis = { (b) Find the projection of b = onto W (c) Find two linearly independent vectors in R* perpendicular to W. Vectors = 1
6. Let W be the set of all vectors of the form W {(a,b,c): a – 2b + 4z = 0} Is W a subspace of the vector space V = R3?