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.b) To show that is a linear transformation, we need to verify the two properties of linearity: additivity and homogeneity. Additivity means that for any vectors and , and homogeneity means that for any scalar and vector . By substituting into these properties, we can show that is indeed a linear transformation.c) To find the matrix representing the linear transformation , we need to determine the coefficients of and in the output vector. In this case, the matrix is: d) To determine if is a linear transformation, we need to check if it satisfies the properties of linearity. In this case, is not a linear transformation because it does not satisfy the homogeneity property.e) To find the value of 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 that makes the system inconsistent is .f) For the given vectors , 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.g) To determine if is a linear transformation, we need to check if it satisfies the properties of linearity. In this case, is a linear transformation.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 , and the dimension is 1.i) To find the basis and dimension of the row space of the given matrix , 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 , and the dimension is 3.j) For the transformation , we need to find the basis for the rank of , the basis for the kernel of , and the rank and kernel of .i) The basis for the rank of is , and the dimension is 2.ii) The basis for the kernel of is , and the dimension is 1.iii) The rank of is 2, and the kernel of is spanned by the vector .