Question

Consider the basis B of R2 consisting of vectors

5
-6
and
-2
-2

Find x in R2 whose coordinate vector relative to the basis B is

[x]B = [-6]
[ 2 ]
and please explain to me how you found it.

Thank You

Answer 1

The idea is to find a linear combination a_1(5, -6) + a_2(-2, -2) = (-6, 2)

It boils down to a system of equations:
Take the augmented matrix:
\[ \left[ \begin {array}{ccc} 5&-2&-6\\ -6&-2&2 \end {array} \right] \]
Reduced form:
\[ \left[ \begin {array}{ccc} 1&0&-{\frac {8}{11}}\\ 0
&1&{\frac {13}{11}}\end {array} \right] \]

-(8/11)*(5, -6) + (13/11)*(-2, -2) = (-6, 2)

Answer 2

the answer is supposed to be another 2 by 1 matrix

Answer 3

\[\left(\begin{matrix}-\frac{8}{11} \\ \frac{13}{11}\end{matrix}\right)\]

Answer 4

Let me try to rewrite it to show you. You are solving a system of equations.

Answer 5

The solution is the column vector (2 by 1 matrix) \[ \left[ \begin {array}{c} -{\frac {8}{11}}\\ {\frac
{13}{11}}\end {array} \right]
\]

Answer 6

\[5x_1 -2x_2 = -6\]\[-6x_1 - 2x_2 = 2\]

Answer 7

Solve for x_1 and x_2 to find the coordinates of the coordinate vector.