 4.2.1: Questions 19 ask for projections onto lines. Also errors e = b  P...
 4.2.2: Questions 19 ask for projections onto lines. Also errors e = b  P...
 4.2.3: Questions 19 ask for projections onto lines. Also errors e = b  P...
 4.2.4: Questions 19 ask for projections onto lines. Also errors e = b  P...
 4.2.5: Questions 19 ask for projections onto lines. Also errors e = b  P...
 4.2.6: Questions 19 ask for projections onto lines. Also errors e = b  P...
 4.2.7: Questions 19 ask for projections onto lines. Also errors e = b  P...
 4.2.8: Questions 19 ask for projections onto lines. Also errors e = b  P...
 4.2.9: Questions 19 ask for projections onto lines. Also errors e = b  P...
 4.2.10: Project a1 = (1,0) onto a2 = (1,2). Then project the result back on...
 4.2.11: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.12: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.13: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.14: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.15: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.16: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.17: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.18: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.19: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.20: Questions 1120 ask for projections, and projection matrices, onto ...
 4.2.21: Questions 2126 show that projection matrices satisfy p2 = P and pT...
 4.2.22: Questions 2126 show that projection matrices satisfy p2 = P and pT...
 4.2.23: Questions 2126 show that projection matrices satisfy p2 = P and pT...
 4.2.24: Questions 2126 show that projection matrices satisfy p2 = P and pT...
 4.2.25: Questions 2126 show that projection matrices satisfy p2 = P and pT...
 4.2.26: Questions 2126 show that projection matrices satisfy p2 = P and pT...
 4.2.27: The important fact that ends the section is this: If AT Ax = 0 then...
 4.2.28: Use pT = P and p2 = P to prove that the length squared of column 2 ...
 4.2.29: If B has rank m (full row rank, independent rows) show that BBT is ...
 4.2.30: (a) Find the projection matrix Pc onto the column space of A (after...
 4.2.31: (a) Find the projection matrix Pc onto the column space of A (after...
 4.2.32: Suppose PI is the projection matrix onto the Idimensional subspace...
 4.2.33: PI and P2 are projections onto subspaces S and T. What is the requi...
 4.2.34: If A has r independent columns and B has r independent rows, AB is ...
Solutions for Chapter 4.2: Projections
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 4.2: Projections
Get Full SolutionsSince 34 problems in chapter 4.2: Projections have been answered, more than 11660 students have viewed full stepbystep solutions from this chapter. Chapter 4.2: Projections includes 34 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.