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Math 130 Linear Algebra
Final
Dec 2006
You may refer to one sheet of notes on this test, and you may use a calculator. You may
leave your answers as expressions such as 8
4e1/3
√2πif you like. Points for each problem are
in square brackets.
Problem 1. [14; 7 points each part] Let Abe the matrix A=
1 5 0 2 −2
0 1 0 3 4
0 0 0 1 7
0 0 0 2 14
.
a. Recall that the row space of an m×nmatrix is the subspace of Rnspanned by the
rows of the matrix. Find a basis for the row space for A.
b. Recall that the null space of a matrix Ais the set of all solutions of the homogeneous
system Ax=0. Find a basis for the null space for A.
Problem 2. [14; 7 points each part] Consider the three vectors u= (1,3,1), v= (4,2,−1),
and w= (−3,1,2).
a. Either prove that the uis in the span of the vectors vand w, or prove that it is not.
b. Are the three vectors u,v, and wlinearly dependent, or linearly independent?
Problem 3. [10] A parallelogram in R3has as adjacent sides the vectors u= ((1,3,−2) and
v= (3,−1,−1). Determine the area of the parallelgram.
Problem 4. [20; 5 points each part] Let Abe the matrix A=
202
021
003
.
a. Write down the characteristic polynomial f(λ) for A.
b. Determine the eigenvalues for A.
c. For each of the eigenvalues of A, find the eigenspace of eigenvectors for that eigenvalue.
d. Is Aa diagonalizable matrix? Explain why or why not.
Problem 5. [20; 10 points each part] Recall that a subset Wof a vector space Vis a
subspace of Vif and only if (1) 0is a vector in W, (2) Wis closed under vector addition,
and (3) Wis closed under scalar multiplication.
a. Prove that the intersection
W1∩W2={v∈V|v∈W1and v∈W2}
1
