Mathematics II TU Board 2081 question paper

Define augmented matrix with an example. Find the general solution of the linear system whose augmented matrix is:

$$\left[\begin{array}{cccc}1& 6& 2& -5\\ -1& 0& 3& 1\\ 0& -1& -2& 3\end{array}\right]$$

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Let $$A=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$$

and Define $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ by $T(x)=Ax$.

Find the image under $$T\left(u=\left[\begin{array}{c}1\\ -3\end{array}\right],v=\left[\begin{array}{c}u\\ u\end{array}\right]\right)$$

• b) Prove that a map $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ defined by $T(x,y)=(x,y)$ is linear.

Define the inverse of a matrix. Find the inverse of matrix $$A=\left[\begin{array}{cc}1& 0\\ 3& 4\end{array}\right]$$ if it exists.

Verify that
$$(AB{)}^{-1}=B^{-1}{A}^{-1}$$
if
$$A=\left[\begin{array}{cc}1& 1\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}2& 1\\ 5& 3\end{array}\right].$$

Find LU factorization of $$A=\left[\begin{array}{cc}2& 5\\ 6& -7\end{array}\right]$$

Compute the determinant by cofactor expansions: $$\left|\begin{array}{cccc}6& 0& 0& 5\\ 1& 7& 2& -5\\ 2& 0& 0& 0\\ 8& 3& 1& 8\end{array}\right|$$

Show that the column vectors $$u=\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right],v=\left[\begin{array}{c}2\\ 1\\ 3\end{array}\right],w=\left[\begin{array}{c}1\\ 2\\ 6\end{array}\right],x=\left[\begin{array}{c}3\\ 1\\ 2\end{array}\right]$$lie in span {u,v,w}{u,v,w}.

Let B={b1,b2}, C={c1,c2} be bases for a vector space V, and suppose $$b_1=6c_1–2c_2,b_2=9c_1–4c_2$$

• a) Find the change of coordinates matrix from BB to CC.
• b) Find $[x{]}_C$.

$$x=-3b_1+2b_2$$

Let $$A=\left[\begin{array}{cc}1& 6\\ 5& 2\end{array}\right]$$ Find the eigenvalues and eigenvectors of AA.

Determine the least squares error in the least-square solution of Ax=b, where $$A=\left[\begin{array}{cc}4& 0\\ 0& 2\\ 1& 1\end{array}\right],x=\left[\begin{array}{c}1\\ 2\end{array}\right],b=\left[\begin{array}{c}2\\ 0\\ 1\end{array}\right]$$

Prove that the binary operation ∗ defined on Zby letting
$$m\ast n=m+n+1$$
is commutative and associative.

Show that $$(\mathbb{Q},+,\cdot )$$ forms a ring.