Question

I need help with those Linear Algebra true or false problems. Please provide a brief explanation if the statement is false.

2. True or False (a) The solution set of the equation Ais a vector space. (b) The rank plus nullity of A equals the number of rows of A (c) The row space of A is equivalent to the column space of AT (d) Every vector in a vector space V can be written as a unit vector. (e) If u-v < 0, then cos θ > 0 where θ is the angle between u, v. (f) Each vector space as only one inner product. e square of the norm of a vector equals the inner product of a vector. (h) The projection of a vector u onto a vector v is a scalar. (i) A set of vectors S is an orthonormal set if each vector is a unit vector and each pair of vectors is orthogonal. G) An orthonormal set of vectors S is a basis for span S.

Answer 1

"(a) We have Ax = 0 Clearly, its solution space forms a vector space hence the given statement is true. (b) False. Because Rank(A) + Nullity (A) = No. of columns. (c) If A = [a_11 a_21 a_n1 a_1 a_22 a_n2 a_1n a-2n a_nn] then A^T = [a_11 a_12 a_1n a_21 a_22 a_2n a_n1 a_n2 a_nn] It is clear that row space of A is the column space of A^T. Hence, true. (d) False, Because every vector can be written as linear combination of unit vectors as unit vectors form basis of vector space but every vector cannot be written as unit vector itself. (e) u middot v < 0 = >