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Unit 1: Coordinate Systems and Matrices - Practice Quiz

ECE183 — Mathematics For Robotics 60 Questions
0 Correct 0 Wrong 60 Left
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1 Which coordinate system represents a point using perpendicular distances along the -, -, and -axes?

Introduction to Coordinate systems Easy
A. Cylindrical coordinate system
B. Spherical coordinate system
C. Polar coordinate system
D. Cartesian coordinate system

2 In a two-dimensional Cartesian coordinate system, the point has an -coordinate of:

Introduction to Coordinate systems Easy
A.
B.
C.
D.

3 What is the origin of a two-dimensional Cartesian coordinate system?

Introduction to Coordinate systems Easy
A.
B.
C.
D.

4 Which coordinate system uses a radial distance and an angle to locate a point in a plane?

Introduction to Coordinate systems Easy
A. Spherical coordinate system
B. Cartesian coordinate system
C. Polar coordinate system
D. Cylindrical coordinate system

5 What does the symbol generally represent?

Differential Length, Area and Volume Easy
A. A differential area
B. A differential volume
C. A differential length
D. A total length

6 What does the symbol represent?

Differential Length, Area and Volume Easy
A. A total area
B. A differential angle
C. A differential volume
D. A differential area

7 What does the symbol represent?

Differential Length, Area and Volume Easy
A. A total volume
B. A differential vector
C. A differential velocity
D. A differential volume

8 In Cartesian coordinates, which expression represents a small volume element?

Differential Length, Area and Volume Easy
A.
B.
C.
D.

9 Which unit is commonly used for an area measurement?

Differential Length, Area and Volume Easy
A.
B.
C.
D.

10 A line integral is evaluated along a:

Line, Surface and Volume Integrals Easy
A. Three-dimensional region
B. Closed surface
C. Curve or path
D. Single coordinate point

11 A surface integral is evaluated over a:

Line, Surface and Volume Integrals Easy
A. Single point
B. Line segment
C. Time interval
D. Surface

12 A volume integral is used to integrate over a:

Line, Surface and Volume Integrals Easy
A. Three-dimensional region
B. Two-dimensional boundary
C. One-dimensional curve
D. Single coordinate axis

13 Which integral is most suitable for finding the total mass distributed throughout a solid with density ?

Line, Surface and Volume Integrals Easy
A.
B.
C.
D.

14 If a constant force acts along a straight displacement of length in the same direction, the work is:

Line, Surface and Volume Integrals Easy
A.
B.
C.
D.

15 What is the main purpose of the differential element in an integral?

Line, Surface and Volume Integrals Easy
A. It identifies the quantity being accumulated
B. It removes the limits of integration
C. It changes a scalar into a matrix
D. It fixes the value of the integrand

16 What is a matrix?

Matrix and its decompositions Easy
A. A single geometric angle
B. A rectangular arrangement of numbers
C. A measurement of volume
D. A curved path in space

17 How many rows and columns does the matrix have?

Matrix and its decompositions Easy
A. 2 rows and 2 columns
B. 3 rows and 3 columns
C. 2 rows and 3 columns
D. 3 rows and 2 columns

18 Which matrix has the same number of rows and columns?

Matrix and its decompositions Easy
A. Rectangular matrix
B. Square matrix
C. Column matrix
D. Row matrix

19 In matrix terminology, what is an element?

Matrix and its decompositions Easy
A. A complete column
B. A complete row
C. An individual entry
D. The matrix dimension

20 In the decomposition , what do and commonly represent?

Matrix and its decompositions Easy
A. Large and uniform matrices
B. Left and unequal matrices
C. Lower and upper triangular matrices
D. Linear and unit matrices

21 A point has cylindrical coordinates . What are its Cartesian coordinates?

Introduction to Coordinate systems Medium
A.
B.
C.
D.

22 The Cartesian point is converted to cylindrical coordinates with . Which result is correct?

Introduction to Coordinate systems Medium
A.
B.
C.
D.

23 A planar vector is rotated counterclockwise through . What are its coordinates after rotation?

Introduction to Coordinate systems Medium
A.
B.
C.
D.

24 Frame is rotated counterclockwise relative to frame , and the origin of is at in frame . If a point has coordinates in frame , what are its coordinates in frame ?

Introduction to Coordinate systems Medium
A.
B.
C.
D.

25 In cylindrical coordinates, a small displacement occurs at with , , and . What is the differential length ?

Differential Length, Area and Volume Medium
A.
B.
C.
D.

26 For the surface , what is the scalar surface-area element at the point ?

Differential Length, Area and Volume Medium
A.
B.
C.
D.

27 At , a cylindrical differential cell has , , and . What is its differential volume?

Differential Length, Area and Volume Medium
A.
B.
C.
D.

28 Using spherical coordinates , where is measured from the positive -axis, find at and when , , and .

Differential Length, Area and Volume Medium
A.
B.
C.
D.

29 Evaluate along the line segment parameterized by for .

Line, Surface and Volume Integrals Medium
A.
B.
C.
D.

30 Find the work done by along the straight path from to .

Line, Surface and Volume Integrals Medium
A.
B.
C.
D.

31 For , calculate the outward flux through the top surface of the cylinder , .

Line, Surface and Volume Integrals Medium
A.
B.
C.
D.

32 Evaluate the surface integral for the part of the plane lying above and .

Line, Surface and Volume Integrals Medium
A.
B.
C.
D.

33 Evaluate over the unit cube .

Line, Surface and Volume Integrals Medium
A.
B.
C.
D.

34 For , which eigenvalue corresponds to the eigenvector ?

Matrix and its decompositions Medium
A.
B.
C.
D.

35 Which pair gives the LU decomposition without pivoting for ?

Matrix and its decompositions Medium
A.
B.
C.
D.

36 Let . In a QR decomposition with positive diagonal entries in , which matrices are correct?

Matrix and its decompositions Medium
A.
B.
C.
D.

37 What are the singular values of , listed from largest to smallest?

Matrix and its decompositions Medium
A.
B.
C.
D.

38 Which lower triangular matrix is the Cholesky factor satisfying for ?

Matrix and its decompositions Medium
A.
B.
C.
D.

39 The least-squares model is fitted to the points , , and . What are the fitted values of and ?

Matrix and its decompositions Medium
A.
B.
C.
D.

40 A planar linear transformation uses . By what factor does it scale areas?

Matrix and its decompositions Medium
A.
B.
C.
D.

41 A point has Cartesian coordinates . Which is its representation in spherical coordinates , where is measured from the positive -axis in the -plane and is measured from the positive -axis?

Introduction to Coordinate systems Hard
A.
B.
C.
D.

42 A robot frame is obtained from frame by a rotation of about , followed by a translation of expressed in frame . What homogeneous transformation maps coordinates from to ?

Introduction to Coordinate systems Hard
A.
B.
C.
D.

43 Let be a vector expressed in frame . Frame is rotated relative to frame by . What are the coordinates of in frame ?

Introduction to Coordinate systems Hard
A.
B.
C.
D.

44 For cylindrical coordinates , a small rectangular coordinate cell has increments , , and . Which expressions give its differential length scales and volume element?

Differential Length, Area and Volume Hard
A. , , ,
B. , , ,
C. , , ,
D. , , ,

45 For the spherical surface , parameterized by with azimuth and polar angle, which oriented differential area vector corresponds to the outward normal?

Differential Length, Area and Volume Hard
A.
B.
C.
D.

46 A surface is parameterized by for and . What is the scalar surface-area element?

Differential Length, Area and Volume Hard
A.
B.
C.
D.

47 In a general orthogonal coordinate system with scale factors , which differential volume element and coordinate-surface area element are correct?

Differential Length, Area and Volume Hard
A. and
B. and
C. and
D. and

48 Evaluate the line integral for along any smooth path from to .

Line, Surface and Volume Integrals Hard
A. The value depends on the chosen path because is not conservative.
B.
C.
D.

49 Evaluate for over the outward-oriented sphere .

Line, Surface and Volume Integrals Hard
A.
B.
C.
D.

50 Let be the upper hemisphere , , with outward normal. For , what is the flux through ?

Line, Surface and Volume Integrals Hard
A.
B.
C.
D.

51 Use the divergence theorem to find the outward flux of through the boundary of the box , , .

Line, Surface and Volume Integrals Hard
A.
B.
C.
D.

52 For the vector field on excluding the -axis, what is the circulation around the circle , traversed counterclockwise once?

Line, Surface and Volume Integrals Hard
A.
B.
C.
D.

53 A scalar density is inside the unit ball. What is its total mass?

Line, Surface and Volume Integrals Hard
A.
B.
C.
D.

54 For , which statement about its eigendecomposition is correct?

Matrix and its decompositions Hard
A. , with orthonormal eigenvectors
B. , with orthonormal eigenvectors
C. , with positive eigenvalues
D. only, because is not diagonalizable over the reals

55 Which is the Cholesky factorization of using a lower triangular factor with positive diagonal?

Matrix and its decompositions Hard
A.
B.
C.
D.

56 For , which statement about its thin QR decomposition is correct?

Matrix and its decompositions Hard
A. has diagonal entries and
B. has diagonal entries and
C. has no QR decomposition because it is rectangular
D. has diagonal entries and

57 The singular values of are:

Matrix and its decompositions Hard
A. and
B. and
C. and
D. and

58 A matrix has rank . Which compact singular-value decomposition correctly represents its rank structure?

Matrix and its decompositions Hard
A. with every diagonal entry of nonzero
B. with and both orthogonal and square
C. , where have orthonormal columns and is positive diagonal
D. for every rectangular matrix, using a diagonal

59 Consider the least-squares problem with full column-rank . Which method is generally numerically preferable to explicitly forming ?

Matrix and its decompositions Hard
A. Solve using a QR factorization of
B. Invert directly even when is rectangular
C. Form and use Gaussian elimination
D. Use the determinant of as the solution vector

60 Let with . Which statement best describes the numerical behavior of solving as ?

Matrix and its decompositions Hard
A. The matrix becomes singular only when exceeds one
B. The matrix becomes ill-conditioned because its determinant approaches zero
C. The matrix becomes better conditioned because all entries remain bounded
D. The matrix becomes orthogonal because its columns become parallel
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Unit 2