Math 2568: Midterm #2
1. Find the Euclidean norm ||x|| of the vector
2. Find a unit vector u that has the same direction as the vector v = i + j + 2k.
3. Find the dot product (i + j + 2k) · (2i − j − k).
4. Find the cross product
5. For the matrix
(a) find the reduced echelon form.
(b) Find a basis for the null space of A
(c) Find a basis for the range of A
(d) Find a basis for the row space of A
6. For the matrix
from the previous problem find the rank and nullity.
7. (True-False-Choice.) Circle the correct answer for each statement.
(1) If S is a subset of a vector space V , then S is a subspace. True False
(2) R2
is a subspace of R3. True False
(3) It is possible to find a pair of two-dimensional subspaces of R3 whose intersection consists of the zero vector 0 only. True False
(4) If x1, x2, . . . , xn span Rn
, then they are always linearly independent. True False
(5) If x1, x2, . . . , xn are linearly independent, then they always span Rn
. True False
(6) If x1, x2, . . . , xn span a vector space V , then they are always linearly independent. True False
(7) If x1, x2, . . . , xn are vectors in a vector space V and
Sp(x1, x2, . . . , xn) = Sp(x1, x2, . . . , xn−1) ,
then x1, x2, . . . , xn are always linearly dependent. True False
(8) If x1, x2, . . . , xn are linearly dependent vectors in a vector space V , then always
Sp(x1, x2, . . . , xn) = Sp(x1, x2, . . . , xn−1) . True False
(9) If A is an m × n matrix, then A and AT always have the same rank. True False
(10) If A is an m × n matrix, then A and AT always have the same nullity. True False
8. Find the coordinates of the vector with respect to the basis
9. (Multiple choice.)
Let A be a 3 × 5 non-zero matrix.
(a) It is known that its rank is one of the following numbers. Circle the correct one.
0, 2, 4, 5, 7, 8, 15.
(b) For the same matrix A as in part (a) circle the value of its nullity.
0, 2, 3, 4, 5, 7, 8.
(c) For the same matrix A as in parts (a) and (b) circle the value of the dimension of its row space.
0, 2, 3, 4, 5, 7, 8.
10. Use Gram-Schmidt process to find an orthogonal basis for the subspace of R
3
spanned by vectors