- 5.1: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.1: In Exercises 1 and 2, examine the four equations and the graphs lab...
- 5.5.2.1: In Exercises 1 and 2, examine the four equations and the graphs lab...
- 5.5.3.1: In Exercises 1 and 2, examine the four equations and the graphs lab...
- 5.2: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.2: In Exercises 1 and 2, examine the four equations and the graphs lab...
- 5.5.2.2: In Exercises 1 and 2, examine the four equations and the graphs lab...
- 5.5.3.2: In Exercises 1 and 2, examine the four equations and the graphs lab...
- 5.3: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.3: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.3: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.3: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.4: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.4: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.4: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.4: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.5: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.5: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.5: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.5: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.6: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.6: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.6: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.6: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.7: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.7: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.7: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.7: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.8: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.8: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.8: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.8: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.9: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.9: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.9: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.9: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.10: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.10: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.10: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.10: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.11: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.11: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.11: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.11: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.12: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
- 5.5.1.12: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.12: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.12: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.13: In Exercises 13 and 14, find the eccentricity. Find the eccentricit...
- 5.5.1.13: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.13: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.13: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.14: In Exercises 13 and 14, find the eccentricity. Find the eccentricit...
- 5.5.1.14: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.14: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.14: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.15: In Exercises 15 to 22, find the equation of the conic section that ...
- 5.5.1.15: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.15: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.15: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.16: In Exercises 15 to 22, find the equation of the conic section that ...
- 5.5.1.16: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.16: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.16: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.17: In Exercises 15 to 22, find the equation of the conic section that ...
- 5.5.1.17: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.17: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.17: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.18: In Exercises 15 to 22, find the equation of the conic section that ...
- 5.5.1.18: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.18: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.18: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.19: In Exercises 15 to 22, find the equation of the conic section that ...
- 5.5.1.19: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.19: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.19: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.20: In Exercises 15 to 22, find the equation of the conic section that ...
- 5.5.1.20: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.20: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.20: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.5.3.21: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.21: In Exercises 15 to 22, find the equation of the conic section that ...
- 5.5.1.21: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.21: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.22: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.22: In Exercises 15 to 22, find the equation of the conic section that ...
- 5.5.1.22: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.22: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.23: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.23: In Exercises 23 to 26, find the equation of the parabola or ellipse...
- 5.5.1.23: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.23: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.24: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.24: In Exercises 23 to 26, find the equation of the parabola or ellipse...
- 5.5.1.24: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.24: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.25: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.25: In Exercises 23 to 26, find the equation of the parabola or ellipse...
- 5.5.1.25: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.25: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.26: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.26: In Exercises 23 to 26, find the equation of the parabola or ellipse...
- 5.5.1.26: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.26: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.27: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.27: Telescope Design The parabolic mirror of a telescope has a concave ...
- 5.5.1.27: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.27: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.28: In Exercises 3 to 28, find the center, vertices, foci, and asymptot...
- 5.28: Arched Door Design The top of an arched door has a semielliptical s...
- 5.5.1.28: In Exercises 3 to 28, find the vertex, focus, and directrix of the ...
- 5.5.2.28: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.29: In Exercises 29 to 34, use the quadratic formula to solve for y in ...
- 5.5.1.29: Find the equation in standard form of the parabola with vertex at t...
- 5.5.2.29: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.30: In Exercises 29 to 34, use the quadratic formula to solve for y in ...
- 5.5.1.30: Find the equation in standard form of the parabola with vertex at t...
- 5.5.2.30: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.31: In Exercises 29 to 34, use the quadratic formula to solve for y in ...
- 5.5.1.31: Find the equation in standard form of the parabola with vertex at (...
- 5.5.2.31: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.32: In Exercises 29 to 34, use the quadratic formula to solve for y in ...
- 5.5.1.32: Find the equation in standard form of the parabola with vertex at (...
- 5.5.2.32: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.33: In Exercises 29 to 34, use the quadratic formula to solve for y in ...
- 5.5.1.33: Find the equation in standard form of the parabola with focus (3, -...
- 5.5.2.33: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.34: In Exercises 29 to 34, use the quadratic formula to solve for y in ...
- 5.5.1.34: Find the equation in standard form of the parabola with focus (-2, ...
- 5.5.2.34: In Exercises 3 to 34, find the center, vertices, and foci of the el...
- 5.5.3.35: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.35: Find the equation in standard form of the parabola that has vertex ...
- 5.5.2.35: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.36: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.36: Find the equation in standard form of the parabola that has vertex ...
- 5.5.2.36: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.37: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.37: Solar Collector Design A mirrored parabolic trough is used to focus...
- 5.5.2.37: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.38: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.38: Ski Design Many contemporary skis have parabolic sidecuts that allo...
- 5.5.2.38: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.39: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.39: Satellite Dish A satellite dish has the shape of a paraboloid. The ...
- 5.5.2.39: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.40: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.40: Radio Telescopes The antenna of a radio telescope is a parab- (0, 0...
- 5.5.2.40: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.41: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.41: Capturing Sound During televised football games, a parabolic microp...
- 5.5.2.41: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.42: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.42: The Lovell Telescope The Lovell Telescope is a radio telescope loca...
- 5.5.2.42: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.43: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.43: The surface area of a paraboloid with radius r and depth d is given...
- 5.5.2.43: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.44: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.44: The Hale Telescope The parabolic mirror in the Hale Telescope at th...
- 5.5.2.44: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.45: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.45: The Lick Telescope The parabolic mirror in the Lick Telescope at th...
- 5.5.2.45: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.46: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.46: Headlight Design A light source is to be placed on the axis of symm...
- 5.5.2.46: In Exercises 35 to 46, find the equation in standard form of each e...
- 5.5.3.47: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.47: Structural Defects Ultrasound is used as a nondestructive method of...
- 5.5.2.47: In Exercises 47 to 54, use the eccentricity of each ellipse to find...
- 5.5.3.48: In Exercises 35 to 48, find the equation in standard form of the hy...
- 5.5.1.48: Fountain Design A fountain in a shopping mall has two parabolic arc...
- 5.5.2.48: In Exercises 47 to 54, use the eccentricity of each ellipse to find...
- 5.5.3.49: In Exercises 49 to 54, use the eccentricity to find the equation in...
- 5.5.1.49: In Exercises 49 to 51, use the following definition of latus rectum...
- 5.5.2.49: In Exercises 47 to 54, use the eccentricity of each ellipse to find...
- 5.5.3.50: In Exercises 49 to 54, use the eccentricity to find the equation in...
- 5.5.1.50: In Exercises 49 to 51, use the following definition of latus rectum...
- 5.5.2.50: In Exercises 47 to 54, use the eccentricity of each ellipse to find...
- 5.5.3.51: In Exercises 49 to 54, use the eccentricity to find the equation in...
- 5.5.1.51: In Exercises 49 to 51, use the following definition of latus rectum...
- 5.5.2.51: In Exercises 47 to 54, use the eccentricity of each ellipse to find...
- 5.5.3.52: In Exercises 49 to 54, use the eccentricity to find the equation in...
- 5.5.1.52: The result of Exercise 51 can be stated as the following theorem: T...
- 5.5.2.52: In Exercises 47 to 54, use the eccentricity of each ellipse to find...
- 5.5.3.53: In Exercises 49 to 54, use the eccentricity to find the equation in...
- 5.5.1.53: The result of Exercise 51 can be stated as the following theorem: T...
- 5.5.2.53: In Exercises 47 to 54, use the eccentricity of each ellipse to find...
- 5.5.3.54: In Exercises 49 to 54, use the eccentricity to find the equation in...
- 5.5.1.54: By using the definition of a parabola, find the equation in standar...
- 5.5.2.54: In Exercises 47 to 54, use the eccentricity of each ellipse to find...
- 5.5.3.55: Loran Two radio transmitters are positioned along the coast, 250 mi...
- 5.5.1.55: Sketch a graph of 4( y - 2) = x x - 1.
- 5.5.2.55: Medicines A lithotripter is an instrument used to remove a kidney s...
- 5.5.3.56: Loran Two radio transmitters are positioned along the coast, 300 mi...
- 5.5.1.56: Find the equation of the directrix of the parabola with the vertex ...
- 5.5.2.56: Construction A circular vent pipe is placed on a roof that has a sl...
- 5.5.3.57: Sonic Booms When a plane exceeds the speed of sound, a sonic boom i...
- 5.5.1.57: Find the equation of the parabola with vertex at the origin and foc...
- 5.5.2.57: The Orbit of Saturn The distance from Saturn to the sun at Saturns ...
- 5.5.3.58: Cooling Tower A vertical cross section of a cooling tower is a port...
- 5.5.1.58: The only information we have about a particular parabola is that (2...
- 5.5.2.58: The Orbit of Venus Venus has a mean distance from the sun of 67.08 ...
- 5.5.3.59: Hyperbolic Gear The following diagram shows a cylindrical worm gear...
- 5.5.2.59: Whispering Gallery An architect wishes to design a large room that ...
- 5.5.3.60: Water Waves If two pebbles are dropped into a pond at different pla...
- 5.5.2.60: Whispering Gallery An architect wishes to design a large room 100 f...
- 5.5.3.61: In Exercises 61 to 68, identify the graph of each equation as a par...
- 5.5.2.61: Halleys Comet Find the equation of the path of Halleys comet in ast...
- 5.5.3.62: In Exercises 61 to 68, identify the graph of each equation as a par...
- 5.5.2.62: Elliptical Pool Table A pool table in the shape of an ellipse has o...
- 5.5.3.63: In Exercises 61 to 68, identify the graph of each equation as a par...
- 5.5.2.63: Bridge Clearance During the 1960s, the London Bridge was dismantled...
- 5.5.3.64: In Exercises 61 to 68, identify the graph of each equation as a par...
- 5.5.2.64: Elliptical Gears The figure on the next page shows two elliptical g...
- 5.5.3.65: In Exercises 61 to 68, identify the graph of each equation as a par...
- 5.5.2.65: Construction A carpenter needs to cut a semielliptical form from a ...
- 5.5.3.66: In Exercises 61 to 68, identify the graph of each equation as a par...
- 5.5.2.66: Orbit of Mars Mars travels around the sun in an elliptical orbit wi...
- 5.5.3.67: In Exercises 61 to 68, identify the graph of each equation as a par...
- 5.5.2.67: In Exercises 67 to 69, use the quadratic formula to solve for y in ...
- 5.5.3.68: In Exercises 61 to 68, identify the graph of each equation as a par...
- 5.5.2.68: In Exercises 67 to 69, use the quadratic formula to solve for y in ...
- 5.5.3.69: In Exercises 69 to 72, use the definition of a hyperbola to find th...
- 5.5.2.69: In Exercises 67 to 69, use the quadratic formula to solve for y in ...
- 5.5.3.70: In Exercises 69 to 72, use the definition of a hyperbola to find th...
- 5.5.2.70: The area A of the ellipse with standard form x2a2 y 2b2 1is given b...
- 5.5.3.71: In Exercises 69 to 72, use the definition of a hyperbola to find th...
- 5.5.2.71: The area A of the ellipse with standard form x2a2 y 2b2 1is given b...
- 5.5.3.72: In Exercises 69 to 72, use the definition of a hyperbola to find th...
- 5.5.2.72: Explain why the graph of is or is not an ellipse. Sketch the graph ...
- 5.5.3.73: Sketch a graph of x x 16 - y y 9 =
- 5.5.2.73: In Exercises 73 to 76, find the equation in standard form of each e...
- 5.5.3.74: Eccentricity Which of the following hyperbolas has the larger eccen...
- 5.5.2.74: In Exercises 73 to 76, find the equation in standard form of each e...
- 5.5.3.75: Telescope Design An astronomer is designing the telescope shown in ...
- 5.5.2.75: In Exercises 73 to 76, find the equation in standard form of each e...
- 5.5.2.76: In Exercises 73 to 76, find the equation in standard form of each e...
- 5.5.2.77: A line segment with endpoints on an ellipse that is perpendicular t...
- 5.5.2.78: A line segment with endpoints on an ellipse that is perpendicular t...
- 5.5.2.79: Show that for any ellipse the length of a latus rectum is 2b2a.
- 5.5.2.80: Use the definition of an ellipse to find the equation of an ellipse...
Solutions for Chapter 5: Topics in Analytic Geometry
Full solutions for College Algebra | 7th Edition
ISBN: 9781439048610
Since 241 problems in chapter 5: Topics in Analytic Geometry have been answered, more than 133368 students have viewed full step-by-step solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9781439048610. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 7. Chapter 5: Topics in Analytic Geometry includes 241 full step-by-step solutions.
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.
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Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
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Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
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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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.
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Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
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Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.
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Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
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Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
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Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
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Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.