Determine if b is a linear combination of a(1), a(2) and a(3). a(1)=(1; -2; 0), a(2)=(0; 1; 2),  a(3)=(5; -6; 8), b=(2; -1; 6).From chapter 1.3, exercise 11 of the book "Linear Algrebra and Its...

Determine if b is a linear combination of a(1), a(2) and a(3).

a(1)=(1; -2; 0), a(2)=(0; 1; 2),  a(3)=(5; -6; 8), b=(2; -1; 6).

From chapter 1.3, exercise 11 of the book "Linear Algrebra and Its Applications" by David C. Lay.

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mathsworkmusic | (Level 2) Educator

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Write the set of equations as an augmented matrix:

 1  0   5  |  2

-2  1  -6  | -1

 0  2   8  |  6

Reduce to echelon form by gaussian elimination

This will be of the form

1  a  b  |  c            or  1  a  b  |  c      or  1  a  b |  c

0  1  d  |  e                0  1   d  |  e          0  0  1 |  d

0  0  1  |  f                 0  0   0 |   0          0  0  0 |  0

Add 2*row 1 to row 2:

1  0  5  |  2

0  1  4  |  3

0  2  8  |  6

Take 2*row 2 from row 3:

1  0  5  |  2

0  1  4  |  3

0  0  0  |  0 

This is now in row echelon form

` ` The coefficient associated with a(3) which we will call `z` can take any value. It is a free parameter. Call the coefficients associated with a(1) and a(2) `x`and `y`respectively.

Then `x + 5z = 2`  and `y + 4z = 3`

`implies x = 2 -5z`  and `y = 3 -4z`

`therefore` `b = (2-5z)a(1) + (3-4z)a(2) + za(3)` for any ` ``z`

So `b` can be written as a linear combination of `a(1), a(2)` and `a(3)` in an infinite number of ways

Yes, b is a linear combination of a(1), a(2) and a(3)

 

 

 

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