If Q = [ūſ ū2), so the columns of Q form an orthonormal basis of V...
7. Claim: Let A be an (n × n) (square) matrix. ·Claim: If A s invertible and AT = A-1 , then the columns of A form an orthonormal basis for R . Claim: If the columns of A form an orthogonal basis for Rn, then A is invertible and A A-1 . Claim: If the columns of A form an orthonormal basis for R", then A is invertible and AT= A-1 . Claim: If the columns of A form...
[-16/257 128. In R3, ū1 12/25 form an orthonormal basis of V = span(ün, ). 3/ 51 (A) Find the 3 x 3 matrix P such that projv (T) = Pi for all i ER (B) Find the 3 x 3 matrix P, such that projv) = P27 for all E R. (C) Use either V or V - to fill in the blanks: ker(projv) = _ im(projv) = _ ker(projy) = __ im(projy) = _
(i) Find an orthonormal basis {~u1, ~u2} for S (ii) Consider the function f : R3 -> R3 that to each vector ~v assigns the vector of S given by f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a linear function. (iii) What is the matrix of f in the standard basis of R3? (iv) What are the null space and the column space of the matrix that you computed in the previous point? Exercise 1. In...
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...
Problem 6. (6 points) (a) Explain why В-Q.(.)}form a basis for R2 forms a basis for R2 (b) Find the coordinate vector of in the basis (c) Suppose the standard matrix of a linear transformation T:R2 R2 is 2-3 Find the matrix of T with respect to the basis B, i.e., find [T]B.
If A is a real matrix with linearly independent columns and A has QR factorization A = QR, then the columns of Q form an orthonormal basis for Col A. O O True False Indicate whether the statement is true or false: if matrix Ais nxn and diagonalizable, then A exists and is diagonalizable. O O True False If u and v are orthonormal vectors with n entries, then u'v = 1. O O True False If vectory is in...
Let w be a subspace of R" and B = {ū1, ... ,üx] be an orthonormal basis for W If we form the matrix U = (ū ū2 - ūk) then the matrix P=UUT is a projection matrix so that Po = Proj, Use the fact that P =P to find all eigenvalues of the matrix P. Hint: Suppose that PŪ = nü for some scalar ܝܠ and non-zero vector Use the fact that p2 = P to find all...
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 =...
linear algebra problem 2. Consider the matrix c=110-3 10-3 10-3 o 10-3 (a) Apply the Gram-Schmidt process to the columns of C, using the standard inner prod- uct. (b) Repeat part (a), this time using 3-digit floating point arithmetic. Is the result an (approximately) orthonormal set? 2. Consider the matrix c=110-3 10-3 10-3 o 10-3 (a) Apply the Gram-Schmidt process to the columns of C, using the standard inner prod- uct. (b) Repeat part (a), this time using 3-digit floating...
Two questions,please! 7. Assume C is a linear code. Prove that G is a generator matrix for C if and only if the columns of G form a basis of C 8. Let V. W U be vector spaces over F of finite dimension and φ: V → W, t : W → U linear maps. Prove that Im(φ)-ker( ) holds if and only if ψφ-0 and dimF1m(φ)-dimF kere). 7. Assume C is a linear code. Prove that G is...