Question

In each case, check that { v1,...vn} is a basis for R^n, and express the given vector b as a linear combination of the basis vectors.

(a). v1=(2,3), v2=(3,5). b=(3,4)

(b) v1=(1,0,3), v2=(1,2,2), v3=(1,3,2). b=(1,1,2)

(c) v1=(1,0,1), v2=(1,1,2), v3=(1,1,1). b=(3,0,1)

Answer 1

              (a). Let A be the matrix with v₁,v₂ and b as columns. Then A =

2	3	3
3	5	4

To determine whether { v₁,v₂} is a basis for R² and whether b can be expressed as a linear combination of v₁,v₂, we will reduce A to its RREF as under:

Multiply the 1st row by ½

Add -3 times the 1st row to the 2nd row

Multiply the 2nd row by 2

Add -3/2 times the 2nd row to the 1st row

Then the RREF of A is

1	0	3
0	1	-1

It implies that { v₁,v₂} is a basis for R² and b = 3v₁-v₂.

        (b). Let A be the matrix with v₁,v₂,v₃ and b as columns. Then A =

1	1	1	1
0	2	3	1
3	2	2	2

To determine whether { v₁,v₂,v₃} is a basis for R³ and whether b can be expressed as a linear combination of v₁,v₂,v₃ we will reduce A to its RREF, which is

1	0	0	0
0	1	0	2
0	0	1	-1

It implies that { v₁,v₂,v₃} is a basis for R³ and b = 0v₁+2v₂-v₃.

         (c ). Let A be the matrix with v₁,v₂,v₃ and b as columns. Then A =

1	1	1	3
0	1	1	0
1	2	1	1

To determine whether { v₁,v₂,v₃} is a basis for R³ and whether b can be expressed as a linear combination of v₁,v₂,v₃ we will reduce A to its RREF, which is

1	0	0	3
0	1	0	-2
0	0	1	2

It implies that { v₁,v₂,v₃} is a basis for R³ and b = 3v₁-2v₂+2v₃.