MA-305 Homework 5 Solutions

Due at 2:35 pm, Tuesday, April 7, 1998


Calculations necessary for these problems may be done either by hand or with Maple. Solutions may be submitted in person in class, or you may email an ASCII text, Maple Text, or Maple Worksheet (.mws) to the TA, John Haws (jchaws@eos.ncsu.edu).

Remember: If you have a question, you may find the answer in the Forum.

  1. Let W=span{(1,1,1),(1,2,-2)}.
    1. Find a basis for the orthogonal complement of W.
    2. Describe the orthogonal complement of W geometrical.

    Solution

    1. An easy way to find the basis is to solve the associated homogeneous system Ax=0, where A is the matrix with the vectors from the basis of W as rows. The vectors associated with the free variables in the parametric solution form a basis for the orthogonal complement. In this case, the basis consists of a single vector:
      { [
      [
      [       		-4
      3
      1       		]
      ]
      ]       		}
    2. Our vectors consist of 3 elements; therefore we are in 3-space. The orthogonal complement is spanned by a single vector. Therefore, the orthogonal complement is a line passing through the origin and perpendicular to the plane W.

  2. For the given matrix, find (i) a basis for its row space, (ii) a basis for its column space, and (iii) its rank.
    1. [
      [
      [
      [       		6
      12
      -3
      9       		4
      8
      -2
      6       		-8
      -14
      6
      -11       		2
      6
      1
      4       		10
      2
      2
      6       		]
      ]
      ]
      ]
    2. [
      [
      [       		1
      1
      -1       		1
      2
      2       		2
      3
      1       		]
      ]
      ]

    Solution

    1. A good choice is the set of non-zero rows from the row echelon form of the matrix.
      (i) { [
      [
      [
      [
      [       		1
      2/3
      0
      5/3
      0       		]
      ]
      ]
      ]
      ]       		, [
      [
      [
      [
      [       		0
      0
      1
      1
      0       		]
      ]
      ]
      ]
      ]       		, [
      [
      [
      [
      [       		0
      0
      0
      0
      1       		]
      ]
      ]
      ]
      ]       		}
      A good choice is the set of basic columns of the original matrix.
      (ii) { [
      [
      [
      [       		6
      12
      -3
      9       		]
      ]
      ]
      ]       		, [
      [
      [
      [       		-8
      -14
      6
      11       		]
      ]
      ]
      ]       		, [
      [
      [
      [       		10
      2
      2
      6       		]
      ]
      ]
      ]       		}
      (iii) The rank is 3.
    2. (i) { [
      [
      [       		1
      0
      1       		]
      ]
      ]       		, [
      [
      [       		1
      1
      0       		]
      ]
      ]       		}
      (ii) { [
      [
      [       		1
      1
      -1       		]
      ]
      ]       		, [
      [
      [       		1
      2
      2       		]
      ]
      ]       		}
      (iii) The rank is 2.

  3. Let u=(1,1,4) and v=(-2,5,1).
    Compute
    1. u·v, the scalar product of u and v.
    2. ||u - v||, the Euclidean norm of u - v.
    3. the angle between u and v.
    4. the projection of u onto v.

    Solution

    1. 7
    2. sqrt(34)
    3. approx. 72.5 degrees
    4. (-7/15, 7/6, 7/30)