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MATH 204
Test 2 April 28, 2006 Name
Put your answers in the space provided. Show your reasoning. The maximum score on the test is 30 points.
- 4 points Let A =
,
1a. Explain why A is singular.
1b. Construct a non-zero, 3 × 3 matrix B for which AB = 0. Note: B may have zero entries but not all the entries in B are zero. Circle your answer.
- 4 points Consider the linear transformation T : R^2 → R^2 which maps e 1 to e 1 − 2 e 2 and maps the vector e 2 to twice e 2. Find the standard matrix of T. Circle your answer.
- 2 points Complete the following definition: A set of vectors is said to be a basis for a subspace H of Rn^ if
- 3 points Solve the equation (A + BX)C = D for X, assuming that A, B, and C are n × n invertible matrices. Circle your answer
- 4 points Find the inverse of the matrix A =
. Show the appropriate steps for computing A− 1
without using your calculator. Circle your answer.
- 2 points Complete the following definition: A set of vectors is said to be a basis for a subspace H of Rn^ if
- 4 points Consider the following five vectors in R^3 :
a 1 =
, a 2 =
, a 3 =
, a 4 =
, b =
.
They are the columns of the following matrix [A b] and the matrix R is its row reduced echelon form:
[A b] =
R =
9a Find a basis for Nul(A). Circle your answer.
9b. Find a basis for Col(A). Rectangle your answer.
