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Material Type: Exam; Professor: Schwede; Class: Linear Algebra; Subject: Mathematics; University: University of Michigan - Ann Arbor; Term: Winter 2007;
Typology: Exams
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Math 217 Winter, 2007 Midterm 1
Problem Possible Points Actual Points 1 10
2 10
3 10
TRUE/FALSE 20
Total 50 In all the problems (with the exception of the TRUE/FALSE), indicate how you arrived at your answer. The answer alone will get at best partial credit.
Problem 1 (10 points)
a) Let v 1 =
, v 2 =
, and v 3 =
. If A = [v 1 v 2 v 3 ],
find the solution set to Ax =
, and describe it geometrically.
b) Are the columns of A linearly independent or dependent?
Problem 3 (10 points) a) Give an example of an invertible 2 × 2 matrix (other than the identity matrix). b) Let T : R^2 → R^2 be the linear transformation given by rotation counter- clockwise by ninety degrees. Write down the standard matrix for T and the inverse of T.
c) Let S : R^2 → R^3 given by the matrix A =
. Let U : R^2 → R^3 be
the composition of S and T , i.e., U (x) = S(T (x)). Write down the standard matrix for U.
TRUE/FALSE (20 points)
Below are ten assertions. For each, circle either T or F to indicate whether you believe the assertion is always TRUE or sometimes FALSE. There is no need to justify your response.
T F Let {v 1 , v 2 ,... , vk} be a set of k linearly independent vectors in R^5. Then k < 6.
T F The span of two vectors is a plane.
T F If A is an n × n matrix such that A = AT^ , then A is invertible.
T F Let u, v, and w be three linearly independent vectors in Rn. Then Span{u, v} 6 = Span{u, v, w}.
T F Let u, v, and w be three linearly dependent vectors in Rn. Then Span{u, v} = Span{u, v, w}.
T F Suppose A and B are invertible matrices of the same size. Then the inverse of (AB)T^ is (AT^ )−^1 (BT^ )−^1.
T F A homogeneous system is always consistent.
T F Suppose v 1 , v 2 ,... , vk are linearly dependent vectors. Then we can write v 1 = c 2 v 2 + · · · + ckvk for some real scalars c 2 ,... , ck.
T F The n×n identity matrix, In, is row equivalent to every invertible n×n matrix.
T F Let T : R^2 → Rm^ be a linear transformation, and let u, v, and w be three vectors in the range of T. Then u, v, w are linearly dependent.
