In Exercises 21 and 22, mark each statement True or False. Justify each answer.a. A subspace of is any set H such that (i) the zero vector is in H; (ii) u; v; and u + v are in H, and (iii) is a scalar and cu is in H.b. If is the same as the column space of the matrix .c. The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of .d. The columns of an invertible n × n matrix form a basis for .e. Row operations do not affect linear dependence relations among the columns of a matrix.

Solution:-Step1To findMark each statement True or False. Justify each answer.a. A subspace of is any set H such that (i) the zero vector is in H; (ii) u; v; and u + v are in H, and (iii) is a scalar and cu is in H.b. If is the same as the column space of the matrix .c. The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of .d. The columns of an invertible n × n matrix form a basis for .e. Row operations do not affect linear dependence relations among the columns of a matrix.Step2a. A subspace of is any set H such that (i) the zero vector is in H; (ii) u; v; and u + v are in H, and (iii) is a scalar and cu is in H.The given statement is true.By the definition of subspace.Step3b. If The given statement is true.As the column space of the matrix consists the which are linearly independent also the span of consists the same vectors.Step4c. The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of .The given statement is false.The set of solutions of a system of m homogeneous equations in n unknowns is a subspace of and not of Step5d. The columns of an invertible n × n matrix form a basis for .The given statement is true.As if a matrix is invertible then it has a non-zero determinant and thus its columns are linearly independent. And since these columns will have n components and being n in total.So , the column of the matrix will form a basis for Step6e. Row operations do not affect linear dependence relations among the columns of a matrix.The given statement is true.As the set of solutions does not change by row operations.