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.

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.

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} ​

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

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

Define subspace of a vector space. Prove that H = \{ \begin{bmatrix} a \\ 0 \\ c \end{bmatrix} : a, c \in \mathbb{R} \right\}​​a0c​​:a,c∈R​ is a subspace of ℝ³.