- #1

karnten07

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**[SOLVED] linear algebra - matrix equations**

## Homework Statement

Consider the matrix A =

1 1 1

2 1 0

1 0 -1

a.)

Show that the equation Ax =

1

2

3

has no solutions in R^3, where R is the set of real numbers.

b.) Find two linearly independant vectors b in R^3 such that the equations Ax=b has a solution in R^3.

## Homework Equations

## The Attempt at a Solution

For part a.) i inserted a vector x of the form

a

b

c

into the equation and then multiplied this x by A. I then had 3 equations in a b and c:

(1) a + b + c = 1

(2) 2a + b =2

(3) a - c = 3

Rearranging (3) i get c = a - 3 and substituting this into (1), it becomes 2a + b = 4

But this contradicts (2) so i conclude that the equation has no solution in R^3.

I am stuck on part b.), if i change the original vector(1,2,3) to b equalling (x,y,z) i can simply replace the terms in the original set of euqations.

In doing this i find that 2a+b=x+z meaning that y = x+z

Is this the right way to do this, and how do i find the numbers in vector b? Could i use x = 1, y=3 and z=2 as one vecotr b as this satisfies my equations? The other being the same numbers but each is negative?

Any help is greatly appreciated, thanks