Page 221 - 35Linear Algebra

11.3 Review Problems 221

Alternatively, this information would often be presented as

   
x x + 2y + 3z
y
L   =  4x + 5y + 6z  .
z 7x + 8y + 9z
You could either rewrite this as
     
x 1 2 3 x
L   =  4 5 6    ,
y
y
z 7 8 9 z
to immediately learn the matrix of L, or taking a more circuitous route:
        
x 1 0 0
y
0
0
L   = L x 0 + y   + z  
  
z 0 1 1
         
1 2 3 1 2 3 x
4
6
5
y
= x   + y   + z   =  4 5 6    .
7 8 9 7 8 9 z
11.3 Review Problems
Reading Problems 1 ,2
Webwork: Basis checks 3,4
Computing column vectors 5,6

2
1. (a) Draw the collection of all unit vectors in R .

1
2
(b) Let S x = , x , where x is a unit vector in R . For which x
0
2
is S x a basis of R ?
3
(c) Sketch all unit vectors in R .

   
 1 0 
3
3
,
0
(d) For which x ∈ R is S x =     , x a basis for R .
1
0 0
 
n
(e) Discuss the generalization of the above to R .
n
2. Let B be the vector space of column vectors with bit entries 0, 1. Write
3
1
2
down every basis for B and B . How many bases are there for B ?
n
4
B ? Can you make a conjecture for the number of bases for B ?
221