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## Homework Statement

1. Do the vectors [tex]j_{1}[/tex]= (1,0,-1,2) and [tex]j_{2}[/tex]= (0,1,1,2) form a basis for the space

*W*= {(

*a,b,c,d*) l

*a*-

*b*+

*c*= 0, -2

*a*- 2

*b*+

*d*=0} ?

2. Find a basis for

*W*= { (

*a,b,c,d*) :

*a*-

*b*+ 2

*d*= 0 , 3

*a*+

*c*+ 3

*d*= 0 }

3. Find the dimension of the following subspaces of [tex]\Re^{3}[/tex]

a) span { (1,0,1),(0,1,1),(2,0,0) }

b) span { (1,0,1),(2,2,4),(2,1,7),(-1,-1-2) }

4. Suppose that a subspace

*W*[tex]\subset[/tex] [tex]\Re^{3}[/tex] has a basis { (1,2,3),(1,0,1) }.

a)Is { (-1,-2,-3), (3,2,5) } a basis for

*W*? Why?

## Homework Equations

How do you do this? Is there a working? or can you solved this by inspection.

## The Attempt at a Solution

1. i substitute each vector in a,b,c,d equation and find the equality, so happen when i substitute in all returns 0, which implies that it does form a basis. Is there a proper working for this, cause i can do this by inspection.

2. I was wondering forming a matrix and row reduced the matrix and get a linear dependence equation and plug in some value for c & d to find a & b.

1 -1 0 2 0 ~ 1 0 1/3 1 0

3 0 1 3 0 0 1 1/3 -1 0

So: a = -1/3c - d & b= d - 1/3c

Let c =1 and d =1

a= -1/3 (1) - 1 = -4/3 & b = 1 - 1/3 (1) = 2/3

so basis (-1/3,2/3,1,1)

3 & 4. No idea what to do.