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Suppose А is an mxn matrix having independent columns and we have the factorization A =...
Suppose that А is an mxn matrix with independent columns and the equation Az = 7...
Suppose that А is an mxn matrix with independent columns and the equation Az = 7 is inconsistent. Then the following statements are true. A. The least squares solution to AT = 5 is given by î = (A” A) "A" 7 B. We can reduce the least squares solution Î = (A" A) "A" 5 as follows. î = (A" A) "A" 5 = = A** (AT) 'A 5 = This calculation follows since when matrices This calculation follows...
Suppose that А is an mxn matrix with independent columns and the equation Añ = is...
Suppose that А is an mxn matrix with independent columns and the equation Añ = is inconsistent. Then the following statements are true. A. The least squares solution to Ax = 6 is given by î = (A” A) A 5 B. We can reduce the least squares solution Î = (A” A)-'A” as follows. î = (A” A)'AT = Â = A-'(AT)-'A" 6 = This calculation follows since when matrices A = QR where Q = (ū ūk) and...
In this exercise, you will work with a QR factorization of an mxn matrix. We will proceed in the ...
In this exercise, you will work with a QR factorization of an mxn matrix. We will proceed in the way that is chosen by MATLAB, which is different from the textbook presentation. An mxn matrix A can be presented as a product of a unitary (or orthogonal) mxm matrix Q and an upper-triangular m × n matrix R, that is, A = Q * R . Theory: a square mxm matrix Q is called unitary (or orthogona) if -,or equivalently,...
If A is a real matrix with linearly independent columns and A has QR factorization A...
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...
#9. Which of the following is not necessarily a valid factorization of the given matrix M?...
#9. Which of the following is not necessarily a valid factorization of the given matrix M? (A) if M is any square matrix, then M = QR, where Q and R are both orthogonal matrices (B) if M has linearly independent columns, then M = QR where Q has orthonormal columns and R is an invertible upper triangular matrix (C) if M is a real symmetric matrix, then M = QDQT for some orthogonal matrix Q and diagonal matrix D...
Suppose an 8 x 10 matrix A has eight pivot columns. Is Col A=R8? Is Nul...
Suppose an 8 x 10 matrix A has eight pivot columns. Is Col A=R8? Is Nul A=R2? Explain your answers. Is Col A =R8? A. Yes. Since A has eight pivot columns, dim Col A is 8. Thus, Col A is an eight-dimensional subspace of R8, so Col A is equal to R8 OB. No, the column space of Ais not R. Since A has eight pivot columns, dim Col A is 0. Thus, Col A is equal to 0....
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be the set of...
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be
the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l
A^T=-A}.
(a) Show that W is a subspace of M2x2(R)
(b) Find a basis for W and determine dim(W).
(c) Suppose T: M2x2(R) is a linear transformation given by
T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You
do not need to verify that T is linear.
3. (17 points)...
Suppose we have a matrix A R. Recall the Golub-Kahan bidiagonalisation pro- cedure and the Lawson-Hanson-Chan (LHC)...
Suppose we have a matrix A R. Recall the Golub-Kahan bidiagonalisation pro- cedure and the Lawson-Hanson-Chan (LHC) bidiagonalisation procedure. Answer the folowing questions: (i) Workout the opcration counts required by the Golub-Kahan bidiagonalisation. (ii) Workout the operation counts required by the LHC bidiagonalisation. (iii) Using the rati derive and explain under what circumstances the LHC is com- putationally more advantageous than the Golub-Kahan. (iv) Suppose we have a bidiagonal matrix B e Rn, show that both B B and BB...
In this exercise you will work with LU factorization of an matrix A. Theory: Any matrix A can be ...
In this exercise you will work with LU factorization of an matrix A. Theory: Any matrix A can be reduced to an echelon form by using only row replacement and row interchanging operations. Row interchanging is almost always necessary for a computer realization because it reduces the round off errors in calculations - this strategy in computer calculation is called partial pivoting, which refers to selecting for a pivot the largest by absolute value entry in a column. The MATLAB...
Suppose we have a matrix A Rmxn. Recall the Golub-Kahan bidiagonalisation pro- cedure and the Lawson-Hanson-Chan...
Suppose we have a matrix A Rmxn. Recall the Golub-Kahan bidiagonalisation pro- cedure and the Lawson-Hanson-Chan (LHC) bidiagonalisation procedure (Section 8.2). Answer the following questions: (i) Workout the operation counts required by the Golub-Kahan bidiagonalisation (ii) Workout the operation counts required by the LHC bidiagonalisation. (iii) Using the ratio m, derive and explain under what circumstances the LHC is com- putationally more advantageous than the Golub-Kahan. we have a bidiagonal matrix B Rnxn, show that both B B and BB...
Suppose that А is an mxn matrix with independent columns and the equation Az = 7 is inconsistent. Then the following statements are true. A. The least squares solution to AT = 5 is given by î = (A” A) "A" 7 B. We can reduce the least squares solution Î = (A" A) "A" 5 as follows. î = (A" A) "A" 5 = = A** (AT) 'A 5 = This calculation follows since when matrices This calculation follows...
Suppose that А is an mxn matrix with independent columns and the equation Añ = is inconsistent. Then the following statements are true. A. The least squares solution to Ax = 6 is given by î = (A” A) A 5 B. We can reduce the least squares solution Î = (A” A)-'A” as follows. î = (A” A)'AT = Â = A-'(AT)-'A" 6 = This calculation follows since when matrices A = QR where Q = (ū ūk) and...
In this exercise, you will work with a QR factorization of an mxn matrix. We will proceed in the way that is chosen by MATLAB, which is different from the textbook presentation. An mxn matrix A can be presented as a product of a unitary (or orthogonal) mxm matrix Q and an upper-triangular m × n matrix R, that is, A = Q * R . Theory: a square mxm matrix Q is called unitary (or orthogona) if -,or equivalently,...
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...
#9. Which of the following is not necessarily a valid factorization of the given matrix M? (A) if M is any square matrix, then M = QR, where Q and R are both orthogonal matrices (B) if M has linearly independent columns, then M = QR where Q has orthonormal columns and R is an invertible upper triangular matrix (C) if M is a real symmetric matrix, then M = QDQT for some orthogonal matrix Q and diagonal matrix D...
Suppose an 8 x 10 matrix A has eight pivot columns. Is Col A=R8? Is Nul A=R2? Explain your answers. Is Col A =R8? A. Yes. Since A has eight pivot columns, dim Col A is 8. Thus, Col A is an eight-dimensional subspace of R8, so Col A is equal to R8 OB. No, the column space of Ais not R. Since A has eight pivot columns, dim Col A is 0. Thus, Col A is equal to 0....
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be
the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l
A^T=-A}.
(a) Show that W is a subspace of M2x2(R)
(b) Find a basis for W and determine dim(W).
(c) Suppose T: M2x2(R) is a linear transformation given by
T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You
do not need to verify that T is linear.
3. (17 points)...
Suppose we have a matrix A R. Recall the Golub-Kahan bidiagonalisation pro- cedure and the Lawson-Hanson-Chan (LHC) bidiagonalisation procedure. Answer the folowing questions: (i) Workout the opcration counts required by the Golub-Kahan bidiagonalisation. (ii) Workout the operation counts required by the LHC bidiagonalisation. (iii) Using the rati derive and explain under what circumstances the LHC is com- putationally more advantageous than the Golub-Kahan. (iv) Suppose we have a bidiagonal matrix B e Rn, show that both B B and BB...
Suppose we have a matrix A Rmxn. Recall the Golub-Kahan bidiagonalisation pro- cedure and the Lawson-Hanson-Chan (LHC) bidiagonalisation procedure (Section 8.2). Answer the following questions: (i) Workout the operation counts required by the Golub-Kahan bidiagonalisation (ii) Workout the operation counts required by the LHC bidiagonalisation. (iii) Using the ratio m, derive and explain under what circumstances the LHC is com- putationally more advantageous than the Golub-Kahan. we have a bidiagonal matrix B Rnxn, show that both B B and BB...
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