Problem 2 a Is in span b If possible write as a lin

Problem 2. (a) Is [ ] in span {[ ],[ ],[ ] } (b) If possible, write [ ] as a linear combination of [ ] , [ ] , [ ]. (c) Is there more than one way to write [ ] as a linear combination of [ ] , [ ], [ ] ? Problem 3. Calculate [ ] . [How does this relate to the previous problem?]

Solution

a) The easiest method is to check the determinant value of the three span vectors, if the determinant is zero, that means they don\'t the span R^3

Determinant

= 1(8 + 12) + 0(0+16) + 5(-4)

= 20 - 20

= 0

Since determinant is zero hence these vector doesn\'t span the complete R^3 space

2 = a + 5c

-1 = 2a + b - 6c

6 = 2b + 8c

Hence it is in the span of the given three vectors

b)

2 = a + 5c

-1 = 2a + b - 6c

6 = 2b + 8c

solving the linear equations we get

Matlab solution

Hence the given vector can be written as a sub-vector

c) Yes there are more solution possible like x=7,y=7 and z=-1

Problem 3

The first matrix is 3X3 and second matrix is 3X1, hence resultant matrix will be 3X1

First element = 1 * 7 + 0 * 7 + 5 * -1 = 2

Second element = -2 * 7 + 1 * 7 + -6 * -1 = -1

Third element = 0 * 7 + 2 * 7 + 8 * -1 = 6

Hence correct answer is [2,-1,6]

The given problem multiplies the given matrix span with the solution to obtained the resultant vector

1	0	5
-2	1	-6
0	2	8