# Linear Algebra

(1) Show that the space R over the field of rational numbers Q with the usual operations is infinite dimensional.

(2) Find the values of lambda and μ so that the system
2x1 + 3x2 + 5x3 = 9
7x1 + 3x2 − 2x3 = 8
2x1 + 3x2 + lamdax3 = μ
has (i) no solution, (ii) a unique solution, and (iii) an infinite number of
solutions.

Hints:::(2) (i) lamda= 5, μ $$\neq$$ 9 (ii) lamda $$\neq$$ 5, μ arbitrary (iii) lamda = 5, μ = 9.

2) Row reduce the augmented matrix to a triangular form (you don't need to get 0s above the diagonal). The last row will be all 0s except for functions of $\mu$ and $\lambda$ in the last two places, corresponding to the equation $f(\mu,\lambda)x_3= g(\mu,\lambda)$. If $f(\mu,\lambda)$ is not 0, then you can divide both sides by $f(\mu,\lambda)$ to get a single value for $x_3$. You will then have to look at the row above to see if the coefficient of $x_2$ is also non-zero. Values of $\mu$ and $\lambda$ that make both of those coefficients non-zero give a unique solution.
If $f(\mu,\lambda)= 0$, then if $g(\mu,\lambda)$ not 0, there is no solution. Values of $\mu$ and $\lambda$ that make f= 0, but g not equal to 0 give no solution.
If $f(\mu,\lambda)$ and $g(\mu,\lambda)$ are both 0, you again need to look at the second row. Values of $\mu$ and $\lambda$ that make the entire row 0 give an infinite number of solutions, values that make the coefficient of $$\displaystyle x_2$$ 0 while the last number in the row non-zero give no solutions.