Math 260 Basic Linear Algebra Midterm Exam II (Fall 2000)

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4 questions on 4 pages Total 120 points

Name and Surname: Student No: Section: Signature: Duration: 110 minutes

1. Suppose that A = [ \begin{matrix}0 & 1 & 1 & 0 \\ 1 & 4 & 2 & 7 \\ 1 + c & 2 + c & 1 + c & 3 + c \\ 1 & 1 & 1 & 1 \end{matrix} ] , where c is a nonzero real

number. Let B be a 4 × 4 matrix with det(B) = 10 and C be the matrix obtained from A by doing the following two row operations: first multiply the fourth row by −1, then add the second row to the fourth row. Compute the following:

a) det (A)

b) det (A^{−1})

c) det (AB)

d) det (BA)

e) det (C)

f) det (C^{t})

g) det (A + \frac{1}{2}A)

h) det (3A)

i) det (−A)

j) det (AA)

2. a) Find the general solution of the following of four linear equations in four unknowns

2x − y + 3z  = 1
−3y + 4z + w = −1
2x − 4y + 7z + w = 0
2x + 5y − 5z − 2w = 3.

b) Using part a), or otherwise, find the nullity and the rank of the linear transformation T : ℝ^{4} → ℝ^{⋭} given by

T (x, y, z, w) = (2x − y + 3z, −3y + 4z + w,
2x − 4y + 7z + w, 2x + 5y − 5z − 2w).

Find a basis for the null space N(T) of T.

3. Let A = [ \begin{matrix}1 & 2 & 3 \\ 0 & −1 & 2 \\ 1 & 0 & 3 \end{matrix} ]

a) Find A^{−1}

b) If A is the matrix of a linear transformation T : ℝ^{⊯} → ℝ^{⊯} relative to the standard basis for ℝ^{⊯}, find the matrix of T relative to the standard basis for the domain and the ordered basis {(3, 2, 3), (2, −1, 0), (1, 0, 1)} for the target space.

4. Each part below is a separate question not related to the others.

a) Find an orthonormal basis for the subspace of ℝ^{⊯} spanned by the three vectors (3, 0, 4), (10, 4, 5), (−4, −4, 3).

b) Let V be the linear space of real polynomials of degree ≤ 3 and let T : V → V be defined by

T (f) = xf′ + f^{2}.

Is T linear? Justify your answer.

c) A linear transformation T : ℝ^{⊯} → ℝ^{⊭} is defined as follows: Each vector in ℝ^{⊯} is first mapped onto ℝ^{⊭} under the mapping (x, y, z) ↦ (x, y), the resulting vector is then rotated anticlockwise 45^{∘} about the origin. Find the matrix representation of T.