## Math 1502 Quiz 5March 21, 2000

### Name: ____________________________

Section: Circle One:
Section A1:Lindsay Bates.Classroom:Skiles 202
Section A2:Eric Forgoston.Classroom:Skiles 246
Section A3:Mohammed Sinnokrot.Classroom:Skilkes 256
Section A4:Kasso Okoudjou.Classroom:Skiles
Section A5:Marcus Sammer. Skiles 140

Open Book and Notes.  Carefully explain your proceedures and answers. Calculators allowed, but answers mush be exact.

### Problem #1 (   points)

Let:

A =

a)(5 points)  Find the column space of this matrx.

b) (5 points) Find the null space of this  matrix

c)(5 points)  Find all solutions to Ax =
.

d) (3 points)  Express the result of c) as a translate   of the null space of A.

##### Ans

Columm space is 2b1 - 3 b2 + b3 = 0.

Or : z = t , y = t, x = -t.  Or the line  t .

c)

Or     =    +    t ,

### Problem 2 (14 points)

a:    (5 points)   Find the  null  space of the matrixc:

b:  (3 points)   Find the dimension of the null space of A

c:  (3 points) Using consertvation of dimension, find the dimension  k of the column space of A.
d (3 points) Find   k linearly inde[pendent coumns of A. (Here k is the dimensoin of the column space of A>)

##### Ans

x  + 2 y + 3 z + 4 w = 0
x +  3 y + 4 z + 5 w =  0
x +  5 y + 6 z + 7 x  =  0

Subract multiples of the 1st from the 2nd and third:
x  + 2 y + 3 z + 4 w = 0
y +    z  +   w = 0
3y + 3 z  + 3 w  = 0

And then:

x  + 2 y + 3 z + 4 w = 0
y +    z  +   w = 0
0  = 0

Set w = t, z = s, solve for the rest of the variables:

y = -s - t,
x + 2 (-s -t) + 3 s + 4 t = 0  or

x = -s -2t

Its two dimensional

Therefore the range (column space) is 4-2 dimensional

The 1st two colums of A are linearly independent.

Converted by Mathematica      March 21, 2000