Mathematics II TU Board 2080 (new course) question paper

What is a system of linear equations ? When the system is consistent ? Find the condition on g, h, k that makes the system consistent.

x_1 – 4x_2 + 7x_3 = g

3x_2 – 5x_3 = h

-2x_1 + 5x_2 – 9x_3 = k

$$LetA=\left[\begin{array}{ccc}1& -5& -7\\ -3& 7& 5\end{array}\right],u=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right],b=\left[\begin{array}{c}-2\\ -2\end{array}\right],$$

and define a transformation T : R^3 → R² by T(x) = Ax then

• a) find T(u).

• b) Find x ∈ R^3 whose image under T is b.

• c) Is x unique ?

Find the least square solution of Ax=b where

$$A=\left[\begin{array}{ccc}1& -3& -3\\ 1& 5& 1\\ 1& 7& 2\end{array}\right],b=\left[\begin{array}{c}5\\ -3\\ -5\end{array}\right]$$

and compute the associated least square error.

Are vectors $$v_1=\left[\begin{array}{c}1\\ 4\\ 0\end{array}\right],v_2=\left[\begin{array}{c}10\\ 2\\ 1\end{array}\right]and{v}_3=\left[\begin{array}{c}-5\\ 0\\ 6\end{array}\right]$$ linearly independent? Justify.

Find LU Factorization. Given the matrix:
$$\left[\begin{array}{ccc}2& 3& 4\\ 4& 5& 10\\ 4& 8& 2\end{array}\right]$$

Compute Det of A where
$$A=\left[\begin{array}{cccc}2& -8& 6& 8\\ 3& -9& 5& 10\\ -3& 0& 1& -2\\ 1& -4& 0& 6\end{array}\right]$$

Show that H = {(a−3b, b−a, a, b) : a, b ∈ R} is a subspace of R^4.

Is $$\left[\begin{array}{c}3\\ 2\end{array}\right]$$ an eigen vector of $$\left[\begin{array}{cc}5& -3\\ -4& 9\end{array}\right]$$ ? If so, find eigenvalue.

Let u = (1, -2, 2, 0). Find a unit vector of v in the same direction of u.

Find the basis and dimension of Nul A where A = \begin{bmatrix} 1 & 2 & 3 & 4 \ 2 & 4 & 7 & 8 \end{bmatrix}.

Define group. Show that (Ζ , .) doesn't form a group.

Show that every field is an integral domain.