Download Math 205B&C Exam 01 - Linear Algebra and more Exams Linear Algebra in PDF only on Docsity!
- Suppose the solutions of a matrix equation Ax = b are written in the form p + vh, where p is
a particular solution of Ax = b and vh gives all solutions of the corresponding homogeneous equation
Ax = 0.
Suppose b =
, p =
and vh = x 2
, where x 2 and x 5 are free.
1a. How many rows does A have? How many columns?
1b. Label the columns of A as c 1 , c 2 ,... , cp. Is the set S = {c 1 , c 2 ,... , cp} linearly independent?
Explain in terms of the definition of linear independence.
1c. Write c 3 as a linear combination of the other columns of A, or explain why this cannot be done.
1d. Write c 4 as a linear combination of the other columns of A, or explain why this cannot be done.
1e. Show how to express b as a linear combination of all p columns of A in such a way that none of the
weights involved are 0.
- Suppose T : R
a → R
z is a transformation. Give the definitions of each of the following:
2a. T is a linear transformation.
2b. T is onto R
z .
2c. Suppose T : R
3 → R
4 is defined by T
x 1
x 2
x 3
x 2 x 3 + 2x 1
0
x 1 + x 2 + x 3
2 x 2 + 7
. Show by example that T is not
a linear transformation and that it actually fails both parts of the definition in (2a).
- Let F =
[
2 − 3 w
1 5 − 2
]
and G =
8 x − 6
7 − 2 1
4 y − 5
, and P =
[
7 23 z
q − 12 9
]
; suppose F G = P.
Show all your work in the following:
4a. Find q.
4b. Find w.
4c. Find z.
4d. Find x and y. Use linear algebra techniques to solve any system this problem requires.
- Suppose that A is a 2 × 2 matrix and the following row operations convert A into I 2 : First, rows 1
and 2 of A are swapped. Then 4 copies of row 1 are subtracted from row 2. Finally, row 2 is multiplied
by 5.
5a. What three elementary matrices E1, E2, and E3 represent these three row operations, respectively?
5b. Use the elementary matrices to find A
− 1 .
5c. Find A.
5d. What is the determinant of A?
5e. Is A singular or nonsingular?
