QUESTION TWO (15 MARKS) Define the following terms as used in transformation i) Rank of a linear transformation. Nullity of a linear transformation b) Show that if T:uarrow v is defined as T(x)=AX ,where A is ntimes m and X is ntimes 1 then I' is a linear transformation. (3 Marks) c) Let Tcdot H_(3)^2arrow 9^2 be a linear transformation defined by T(x_(1),x_(2))=(x_(1)+2x,,3x_(1)-x_(2)) Find the matrix M representing T. (3 marks) d) Determine if I defined as T:H_(3)^-arrow H_(2)^2 defined as T(x_(1),x_(2),x_(1))=(2x_(1)-x_(2),x_(1)-x_(1)+1) is a linear transformation. (4 Marks). c) Find hsuch that [} 2&h&4&4 3&6&:&7 ] is the augmented matrix of an inconsistent system. (3Marks) (} 2&4&4&3 3&6&:7 ) SECTION B-ATTEMPT ANY THREE QUESTIONS IN TIIIS SECTION QUESTION THREE (13 MARKS) a) Consider the vectors (1,4),(2,3),(3,2) . Are these vectors linearly independent? (3 Marks) b) Determine if T defined as T:9R^3arrow 9^2 delined as T(x_(1),x_(2),x_(3))=(x_(1)+x_(3),2x_(2)-x_(3)) is a linear transformation. (4 Marks) c) Find the basis and dimension of the solution space for the equations x_(1)+x_(2)-x_(3)=0 -2x_(1)-x_(2)+2x_(3)=0 -x_(1)+x_(3)=0 (6 marks) QUESTION FOUR (13 MARKS) a) Find the basis and dimension of the row space of the matrix A=[} 2&-1&3 1&1&5 -1&2&2 ] (3 Marks) b) The transformation T:9^3arrow 9^3 is defined as T(x)=[} 2&0&-1 4&0&-2 0&0&0 ] Find i) Basis for rank of T. ii) Basis for Kerhel of T ii) (4 Marks) Rank of T and Kernel of T (4 Marks) (2 Marks) (1 Mark)

a) The rank of a linear transformation is the dimension of the image of the transformation. It represents the maximum number of linearly independent columns in the matrix representation of the transformation.<br /><br />b) To show that $T(x) = AX$ is a linear transformation, we need to verify the two properties of linearity: additivity and homogeneity. Additivity means that $T(u + v) = T(u) + T(v)$ for any vectors $u$ and $v$, and homogeneity means that $T(\alpha u) = \alpha T(u)$ for any scalar $\alpha$ and vector $u$. By substituting $T(x) = AX$ into these properties, we can show that $T$ is indeed a linear transformation.<br /><br />c) To find the matrix $M$ representing the linear transformation $T(x_{1},x_{2}) = (x_{1} + 2x_{2}, 3x_{1} - x_{2})$, we need to determine the coefficients of $x_{1}$ and $x_{2}$ in the output vector. In this case, the matrix $M$ is:<br />\[ M = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix} \]<br /><br />d) To determine if $T(x_{1},x_{2},x_{3}) = (2x_{1} - x_{2}, x_{1} - x_{1} + 1)$ is a linear transformation, we need to check if it satisfies the properties of linearity. In this case, $T$ is not a linear transformation because it does not satisfy the homogeneity property.<br /><br />e) To find the value of $h$ such that the augmented matrix is inconsistent, we need to perform row reduction to obtain the row-echelon form. After performing row reduction, we find that the value of $h$ that makes the system inconsistent is $h = 3$.<br /><br />f) For the given vectors $\{(1,4),(2,3),(3,2)\}$, we need to check if they are linearly independent. By forming a matrix with these vectors and performing row reduction, we find that the vectors are linearly independent.<br /><br />g) To determine if $T(x_{1},x_{2},x_{3}) = (x_{1} + x_{3}, 2x_{2} - x_{3})$ is a linear transformation, we need to check if it satisfies the properties of linearity. In this case, $T$ is a linear transformation.<br /><br />h) To find the basis and dimension of the solution space for the given system of equations, we need to perform row reduction to obtain the row-echelon form. After performing row reduction, we find that the basis for the solution space is $\{(1,1,1)\}$, and the dimension is 1.<br /><br />i) To find the basis and dimension of the row space of the given matrix $A$, we need to perform row reduction to obtain the row-echelon form. After performing row reduction, we find that the basis for the row space is $\{(2,-1,3),(1,1,5),(-1,2,2)\}$, and the dimension is 3.<br /><br />j) For the transformation $T(x) = [\begin{matrix} 2&0&-1\\ 4&0&-2\\ 0&0&0\end{matrix} ][\begin{matrix} x_{1}\\ x_{2}\\ x_{3}\end{matrix} ]$, we need to find the basis for the rank of $T$, the basis for the kernel of $T$, and the rank and kernel of $T$.<br /><br />i) The basis for the rank of $T$ is $\{(2,4),(0,-2)\}$, and the dimension is 2.<br /><br />ii) The basis for the kernel of $T$ is $\{(1,0,1)\}$, and the dimension is 1.<br /><br />iii) The rank of $T$ is 2, and the kernel of $T$ is spanned by the vector $(1,0,1)$.