Let w be a subspace of R" and B = {ū1, ... ,üx] be an orthonormal...
Suppose that {ūj, ..., ūk} is an orthonormal basis for a subspace W of R" and we form the matrix U = (ū; ū2 ... ük) Then the matrix P= UUT has the property that p2 = P . This follows for the following reason(s). A. We know that P= I and so P2 = 1? = I = P B. We can calculate p2 = (UUT) (UUT) = U (UTU) UT = UIUT C. Since P is a projection...
Suppose that {ū1, ... , ūk} is a basis for a subspace W of R" and that the vector Ū E span{ū1, ... , ūk}. Then û = Proj, Ū = ū. True O False Suppose that W is a subspace of R" and that the vector ŪER" .Then if û = Projű we have Ilu - Oll < 110 - ūll for all vectors ū EW . That is, <- is the vector in W that is closest to...
If an пXp matrix U has orthonormal columns, then UUT= for all TER" True False Let w be a subspace of R" Suppose that P and Q are nxn matrices so that Po = Proj, and Qü = Proj, for all vectors U ER" then P+Q = 1 Hint: Every vector ÜER" can be written uniquely as the sum of a vector in w and a vector in Qu = Proj, 1 for all vectors ŪER" , then P+Q =...
Exercise 6 (6.4.35, p.452) Let A e Cnxn, and let S be a k-dimensional subspace of C". Then a vector ve S is called a Ritz vector of A from S if and only if there is a pie C such that the Rayleigh-Ritz-Galerkin condition Av – uv Is holds, that is, (Av – uv, s) = 0 for all s E S. The scalar u is called the Ritz value of A associated with v. Let 91, ...,qk be...
#8. Let W be the subspace of R3 spanned by the two linearly independent vectors v1 = (-1,2,2) and v2 = (3, -3,0). (a) Use the Gram-Schmidt orthogonalization process to find an orthonormal basis for W. (b) Use part (a) to find the matrix M of the orthogonal projection P: R W . (c) Given that im(P) = W, what is rank(M)?
101-2019-3-b (1).pdf-Adobe Acrobat Reader DC Eile Edit iew Window Help Home Tools 101-2019-3-b (1) Sign In x Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y, x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V -> V such that U is not an...
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,...
5. Let (a) (2 marks) Find all eigenvalues of A (b) (4 marks) Find an orthonormal basis for each eigenspace of A (you may find an orthonormal basis by inspection or use the Gram-Schmidt algorithm on each eigenspace) (c) (2 marks) Deduce that A is orthogonally diagonalizable. Write down an orthogonal matrix P and a diagonal matrix D such that D P-AP. (d) (1 mark) Use the fact that P is an orthogonal matrix to find P-1 (e) (2 marks)...
Hi, can you please solve this and show work. Let W be a 2-dimensional subspace of R'. Recall that the function T:X → projw X, mapping any vector to its projection onto W is a linear transformation. Let A be the standard matrix of T. a) Explain why Ax = x for any vector x in W. Show that Null(A) = Wt. What is dim(Null(A))?| (Hint: Recall that, for any vector x, X - projw x is orthogonal to W.)...
0/1 pts Inooreat Question 9 Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why WR Two subspaces are the same when one subspace is a subset of the other subspace. Two subspaces are the same when they are spanned by the same vectors Two subspaces are the same when they are subsets of the same space Two subspaces are the same when they have the same dimension Incorrect 0/1 pts Question 10 Let U...