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Subject

Mathematics II

The course contains concepts and techniques of linear algebra. The course topics include systems of linear equations, determinants, vectors and vector spaces, eigenvalues and eigenvectors and singular value decomposition of matrix.

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Define subspace of a vector space. Prove that H = $\{ \begin{bmatrix} a \\ 0 \\ c \end{bmatrix} : a, c \in \mathbb{R} \right\}$ is a subspace of ℝ³.

Not Answered Mathematics II Asked on 2082 Exam

Let A = $\begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix}$ . Find 5|A| − 2|I|. Is det(5A) equal to 5 det A? Justify.

Not Answered Mathematics II Asked on 2082 Exam

Let A = $\begin{bmatrix} 1 & 2 \\ 3 & 0 \end{bmatrix}$ . Find A³.

Not Answered Mathematics II Asked on 2082 Exam

a) Find the basis and dimension of the subspace H = $\begin{bmatrix} a-3b+6c \\ 5a+4d \\ b-2c-d \\ 5d \end{bmatrix} :a,b,c,d∈R$

(b) What is rank of a matrix? Find the rank of a matrix A = $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{bmatrix}$

Not Answered Mathematics II Asked on 2082 Exam

Let \( T \) be defined by \( T(x) = Ax \), where
\[ A = \begin{bmatrix} 1 & -5 & -7 \\ -3 & 7 & 5 \end{bmatrix} \]
Find a vector \( x \) whose image under \( T \) is \( b \), where
\[ b = \begin{bmatrix} -2 \\ -2 \end{bmatrix} \]
and determine whether \( x \) is unique or not.

Not Answered Mathematics II Asked on 2082 Exam

Define homogeneous linear system of equations with an example. Which type of homogeneous equation has a nontrivial solution? Determine the value of x, y, and z if the system of equation x − 2y + 3z = 0, −x + 2y − 4z = 0, 2x − 4y + 9z = 0 has a nontrivial solution.

Not Answered Mathematics II Asked on 2082 Exam

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

Not Answered Mathematics II Asked on 2081 Exam

Prove that the binary operation defined on by letting
\[
m \ast n = m + n + 1
\]
is commutative and associative.

Not Answered Mathematics II Asked on 2081 Exam

Determine the least squares error in the least-square solution of , where \[
A = \begin{bmatrix} 4 & 0 \\ 0 & 2 \\ 1 & 1 \end{bmatrix}, \quad
x = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad
b = \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}
\]

Not Answered Mathematics II Asked on 2081 Exam

Let \[
A = \begin{bmatrix} 1 & 6 \\ 5 & 2 \end{bmatrix}
\] Find the eigenvalues and eigenvectors of $A$ .

Not Answered Mathematics II Asked on 2081 Exam

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