Discrete Mathematical Structures by Pearson

12-28 Discrete Mathematical Structures

As no zero row occurs in the echelon form of the matrix, the set of vectors

 

xxx

123

,,

is

linearly independent.

(2) If the given vector lies in the subspace spanned by the vectors

 

xxx

123

,, then there will

exist scalars c

1

, c

2

, c

3

not all zero such that

(4, -5, 9, -7) = c

1

(-1, 2, 5, 2) + c

2

(3, 0, 4, -1) + c

3

(1, 1, -2, 1)

This gives the following system of equations:

-c

1

+ 3c

2

+ c

3

= 4 (i)

2c

1

+ c

3

= -5 (ii)

5c

1

+ 4c

2

- 2c

3

= 9 (iii)

2c

1

- c

2

+ c

3

= -7 (iv)

Its solution is c

1

= -1, c

2

= 2, c

3

= -3, which shows that the vector (4, -5, 9, -7) lies in the

subspace of R

4

spanned by the given vectors. ...