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CSC 305 Study Questions for Vectors, Matrices, and Transformations
创建日期：2024-06-24 11:42:30
标签： CSC 305
Matrices, and Transformations

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CSC 305 Study Questions for Vectors, Matrices, and Transformations


These questions will attempt to build your intuition about transformations, coordinate systems, and basic geometry used in this course. Most of the questions are straightforward, with brief answers and no tricks. In some cases, many answers will be correct, but you can probably guess what answer is most appropriate by looking at the lecture slides. Answers will not be provided.

  1. What is a scalar?
  2. What is a vector?
  3. What is a matrix?
  4. What is a vector, in matrix terms?
  5. How do you compute the magnitude of a vector?
  6. How do you compute the norm of a vector?
  7. How do you compute the length of a vector?
  8. What is a unit vector?
  9. How do you add two vectors?

10.How do you multiply a scalar and a vector?

11.What are the basic properties of adding vectors and vector-scalar multiplication?

12.What is the geometric interpretation of adding two vectors?

13.What is the geometric interpretation of subtracting two vectors?

14.What is the geometric interpretation of a scalar-vector multiplication?

15.What is the geometric interpretation of normalizing a vector?

16.What is a linear combination of vectors?

17.What is an affine combination of vectors?

18.What does it mean for a collection of vectors to be linearly independent of each other?

19.What is the geometric interpretation of linearly independent vectors?

20.What is a vector space?

21.What is a generator set?

22.What is a basis (base) of a vector space?

23.What is the difference between a generator set and a basis?

24.What is the typical meaning of i, j,k or i1, i2, i3, . . . in?

25.What is the difference between a right-handed 3-d coordinate system and a left-handed 3-d coordinate system?

26.How can we represent any vector in terms of i1, i2, i3, . . . in?

27.How do you compute a dot product?

28.What type of objects are the inputs/output of a dot product? In other words, if you implemented afunction named dotProduct(…), what data type are the arguments, and what data type is the return value? (e.g. vector, scalar or matrix)

29.What are the basic properties of dot products?

30.What is interesting about the dot product of a vector with itself?

31.What is the relationship between the angle between two vectors and their dot product?

32.What is special about this relationship (in the previous question) when both vectors are unit length?

33.How can you tell if two vectors are perpendicular?

34.How can you tell if the angle between two vectors is acute (less than perpendicular)?

35.How can you tell if the angle between two vectors is obtuse (more than perpendicular)?

36.What does it mean for a vector to be normal to another vector?

37.What does it mean for a vector to be orthogonal to another vector?

38.What does it mean for a vector to be normal to a surface?

39.What is a cross product?

40.What type of objects are the inputs/output of a cross product? In other words, if you implemented a function named crossProduct(…), what data-type are the arguments, and what data type is the return value? (e.g. vector, scalar or matrix)

41.What are basic properties of a cross product?

42.What is the geometric interpretation of a cross product?

43.Given C = A × B (the cross product of A and B), what is the relationship between C and B? between C and A? between A and B? (one of those 3 combinations is a trick question)

44.What is a point?

45.What is the distinction between a point and a vector?

46.What is the geometric interpretation of subtracting two points?

47.What is the geometric interpretation of adding a point and a vector?

48.What is the homogeneous representation of a vector?

49.What is the homogeneous representation of a point?

50.What is a linear combination of points?

51.What is an affine combination of points?

52.What is the geometric interpretation of adding two points (in homogeneous coordinates)?

53.What is the geometric interpretation of a linear combination of two points?

54.What is the geometric interpretation of an affine combination of two points?

55.What is a symmetric matrix?

56.What is an identity matrix?

57.How do you add two matrices?

58.What are the basic properties of adding matrices?59.How do you multiply two matrices?

60.What are the basic properties of multiplying matrices?

61.How do you multiply a matrix and a vector?

62.Can you name two interesting geometric interpretations of matrix-vector multiplications? They were seen in class but not explicitly pointed out.

63.Can you reverse the order of addition for matrices? (does A +B = B +A?)

64.Can you reverse the order of multiplication for matrices? (does AB = BA?)

65.What is the inverse of AB, where A and B are matrices?

66.What is the matrix form of a dot product?

67.What are the three ways to describe a line?

68.What are the three ways to describe a plane?

69.What is a transformation?

70.What type of object is commonly used to represent a transformation (scalar, matrix, or vector)?

71.What is the main property of affine transformations?

72.What are the four elementary affine transformations?

73.What is the matrix form of a translation by Tx, Ty , and Tz?

74.What is the matrix form of a scaling by Sx, Sy , and Sz?

75.What is the matrix form of rotation around the origin by θ (in 2-d)?

76.What is the matrix form of a rotation around the z-axis by θ?

77.What is the matrix form of a rotation around the y-axis by θ?

78.What is the matrix form of a rotation around the z-axis by θ?

79.What is the matrix form of a shear in x (in 2-d)?

80.What are the inverses of for the matrices in the previous 7 questions?

81.What is interesting to note about rotation about the y-axis versus the x-axis and z-axis?

82.Given a pure rotation, what is interesting about its inverse?

83.With our notation using column vectors, let M1 and M2 be transformations. If we write M1M2v,which transform is being applied first to v? Is it possible to perform operations in a different order?

84.If M1 and M2 are translations, does M1M2= M2 M1 ?

85.If M1 and M2 are rotations, does M1 M2 = M2M1? (think carefully on this one…)

86.If M1 is a translation and M2 is a rotation, does M1 M2 = M2M1?

87.What are the series of elementary transforms you would use to reflect a point about the x-axis?

88.What are the series of elementary transforms you would use to rotate about the point (4, 5)?

89.What are the series of elementary transforms you would use to scale around the point (4, 5)?

90.What are the series of elementary transforms you would use to reflect a point about the line y = ax?91.What are the series of elementary transforms you would use to rotate a point about an arbitrary vector that passes through the origin?

92.What are the properties of affine transforms?

93.What information is needed to fully specify a coordinate system?

94.Given two coordinate systems, CS1 and CS2 , if CS2 is defined with respect to CS1, what is the rotation part of the matrix that transforms a point in CS2 to a point in CS1? (hint: the answer is explicitly in the lecture slides somewhere). What is the full matrix?

95.What are two interesting geometric interpretations of transformations?


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