Calculus and Linear Algebra Math for Computer Science practice questions and solutions
Matrices and Transformations,
1. Which of the following sets of vectors span R3?
,1).
(b) (1, , , ,−2), and (1,3,1).
(c) (−1,2,3), (2,1,−1), and (4,7,3).
(d) (1,0,2), (0,1,0), (−1,3,0), and (1,−4,1).
2. Which of the following sets of vectors span P3 = {at3 + bt2 + ct + d}?
(a) t + 1, t2 − t, and t3.
(b) t3 + t and t2 + 1.
(c) t2 + t + 1, t + 1, 1, and t3.
(d) t3 + t2, t2 − t, 2t + 4, and t3 + 2t2 + t + 4.
3. Are the following sets of vectors linearly dependent or independent? If they are
dependent, write one as a linear combination of the
others.
(a) (13.
(b) (− , , , ,− ,7,3) in R3.
(c) (1,2), (2,3), and (8,−2) in R2.
(d) t2 + 2t + 1, t3 − t2, t3 + 1, and t3 + t + 1 in P3.
4. What is the dimension of the following spaces?
(a) The set of 2 × 2 symmetric matrices, A = AT.
(b) The set of 2 × 2 matrices
,
with a + d = 0.
(c) The set {(x,y,x − 3y,2y − x) | x,y ∈ R} inside of R4. 5. What is the column space
and row space of the matrix
?
6. Find an (infinite) basis for the space of all polynomials
P = {anxn + zn−1xn−1 + ··· + a1x + a0 | for all n}.
,2 PROBLEM SETS 15–17
7. Suppose {v1,...,vn} spans a vector space V , and suppose that vn is a linear combination of
v1 through vn−1. Then show that {v1,...,vn−1} spans V as well.
8. If A is a 4 × 6 matrix, show that the columns of A are linearly dependent.
1
9. Compute
,
for n = 3, 5 and 100 using methods from recitation.
, PROBLEM SETS 15–17 3
Problem Set 16: Due 27 September 2000.
1. Check that (1,1,0), (0,1,1), and (1,0,1) form a basis for R3. Transform this basis into an
orthonormal basis using the Gram-Schmidt algorithm. Check that the resulting vecotrs
are indeed orthogonal!
2. Check that the vectors (1,1,1,1), (1,1,1,0), (1,1,0,0) and (1,0,0,0) form a basis of 4. Use
the Gram-Schmidt algorithm to make this into an orthogonal basis. Don’t R
worry about normalizing this basis – the result is already a little messy! 3.
Consider the orthonormal vectors
&
in R3. Find some other vector
such that v1, v2, and b are a basis of R3. Then use the Gram-Schmidt algorithm to make
your basis into an orthogonal one.
4. Check that the matrix
is an orthogonal matrix by checking that Q · QT = I. Also, check that ||Q · v|| = ||v||
for the vector .
5. We have the following theorem.
Theorem 1. If {v1,...,vn} is an orthonormal basis for Rn, then for any v ∈ Rn, we can write v
= c1v1 + ···cnvn,
where ci = v · vi (1 ≤ i ≤ n), where · is the dot product.
(a) Use this theorem to write the vector (3,2) as linear combinations of the vectors
&
(b) Use this theorem to write (1,2,−1) in terms of the basis
&
(c) (Optional) Prove the above theorem.
6. Consider the two bases of R3,
Matrices and Transformations,
1. Which of the following sets of vectors span R3?
,1).
(b) (1, , , ,−2), and (1,3,1).
(c) (−1,2,3), (2,1,−1), and (4,7,3).
(d) (1,0,2), (0,1,0), (−1,3,0), and (1,−4,1).
2. Which of the following sets of vectors span P3 = {at3 + bt2 + ct + d}?
(a) t + 1, t2 − t, and t3.
(b) t3 + t and t2 + 1.
(c) t2 + t + 1, t + 1, 1, and t3.
(d) t3 + t2, t2 − t, 2t + 4, and t3 + 2t2 + t + 4.
3. Are the following sets of vectors linearly dependent or independent? If they are
dependent, write one as a linear combination of the
others.
(a) (13.
(b) (− , , , ,− ,7,3) in R3.
(c) (1,2), (2,3), and (8,−2) in R2.
(d) t2 + 2t + 1, t3 − t2, t3 + 1, and t3 + t + 1 in P3.
4. What is the dimension of the following spaces?
(a) The set of 2 × 2 symmetric matrices, A = AT.
(b) The set of 2 × 2 matrices
,
with a + d = 0.
(c) The set {(x,y,x − 3y,2y − x) | x,y ∈ R} inside of R4. 5. What is the column space
and row space of the matrix
?
6. Find an (infinite) basis for the space of all polynomials
P = {anxn + zn−1xn−1 + ··· + a1x + a0 | for all n}.
,2 PROBLEM SETS 15–17
7. Suppose {v1,...,vn} spans a vector space V , and suppose that vn is a linear combination of
v1 through vn−1. Then show that {v1,...,vn−1} spans V as well.
8. If A is a 4 × 6 matrix, show that the columns of A are linearly dependent.
1
9. Compute
,
for n = 3, 5 and 100 using methods from recitation.
, PROBLEM SETS 15–17 3
Problem Set 16: Due 27 September 2000.
1. Check that (1,1,0), (0,1,1), and (1,0,1) form a basis for R3. Transform this basis into an
orthonormal basis using the Gram-Schmidt algorithm. Check that the resulting vecotrs
are indeed orthogonal!
2. Check that the vectors (1,1,1,1), (1,1,1,0), (1,1,0,0) and (1,0,0,0) form a basis of 4. Use
the Gram-Schmidt algorithm to make this into an orthogonal basis. Don’t R
worry about normalizing this basis – the result is already a little messy! 3.
Consider the orthonormal vectors
&
in R3. Find some other vector
such that v1, v2, and b are a basis of R3. Then use the Gram-Schmidt algorithm to make
your basis into an orthogonal one.
4. Check that the matrix
is an orthogonal matrix by checking that Q · QT = I. Also, check that ||Q · v|| = ||v||
for the vector .
5. We have the following theorem.
Theorem 1. If {v1,...,vn} is an orthonormal basis for Rn, then for any v ∈ Rn, we can write v
= c1v1 + ···cnvn,
where ci = v · vi (1 ≤ i ≤ n), where · is the dot product.
(a) Use this theorem to write the vector (3,2) as linear combinations of the vectors
&
(b) Use this theorem to write (1,2,−1) in terms of the basis
&
(c) (Optional) Prove the above theorem.
6. Consider the two bases of R3,
