- 10.3.1: Find the volume of each solid named in Exercises 16. All measuremen...
- 10.3.2: Find the volume of each solid named in Exercises 16. All measuremen...
- 10.3.3: Find the volume of each solid named in Exercises 16. All measuremen...
- 10.3.4: Find the volume of each solid named in Exercises 16. All measuremen...
- 10.3.5: Find the volume of each solid named in Exercises 16. All measuremen...
- 10.3.6: Find the volume of each solid named in Exercises 16. All measuremen...
- 10.3.7: In Exercises 79, express the total volume of each solid with the he...
- 10.3.8: In Exercises 79, express the total volume of each solid with the he...
- 10.3.9: In Exercises 79, express the total volume of each solid with the he...
- 10.3.10: Use the information about the base and height each solid to find th...
- 10.3.11: Sketch and label a square pyramid with height H feet and each side ...
- 10.3.12: Sketch and label two different circular cones, each with a volume 2...
- 10.3.13: Mount Fuji, the active volcano in Honshu, Japan, is 3776 m high and...
- 10.3.14: Bretislav has designed a crystal glass sculpture. Part of the piece...
- 10.3.15: Jamala has designed a container that she claims will hold 50 in3 . ...
- 10.3.16: Find the volume of the solid formed by rotating the shaded figure a...
- 10.3.17: Find the volume of the liquid in this right rectangular prism. All ...
- 10.3.18: Application A swimming pool is in the shape of this prism. A cubic ...
- 10.3.19: Application A landscape architect is building a stone retaining wal...
- 10.3.20: As bad as tanker oil spills are, they are only about 12% the 3.5 mi...
- 10.3.21: ind the surface area of each of the following polyhedrons. (See the...
- 10.3.22: Given the triangle at right, reflect D over AC to D. Then reflect D...
- 10.3.23: In each diagram, WXYZ is a parallelogram. Find the coordinates of Y
Solutions for Chapter 10.3: Volume of Pyramids and Cones
Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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!).
Gram-Schmidt 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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).