1 Write your solutions in complete grammatically correct sen

1. Write your solutions in complete, grammatically correct sentences. Furthermore, make sure that words in your sentences agree with each other mathematically. For example, the phrase \"the matrix A is linearly independent\" does not mean anything linear independence is a property of sets of vectors. Instead, one may write \"the set of columns of A is linearly independent\" or \"{At is linearly independent\", the latter means A is a nonzero matrix (why?). Another example would be 1,2, -1 is 3\"; what is, probably, meant is \"the size of 1, 2, -1 is 3\" John was solving the following problem \"Prove that the dimension of the column space of a matrix A is equal to the number of leading 1s in the reduced row echelon form of A.\" His answer was: \"Let A be a matrix. The column space of A is the pivot columns of A proven in class), and therefore is equal to the number of the leading 1s in the RREF of A Write the corrected solution here. (A marks

Solution

For a zero matrix, all its columns are zero vectots so that the associated linear transformation is not one-to one. Thus, when we say that the columns of a matrix A are linearly dependent, it implies that A is a non-zero matrix. The column space of a matrix A is the span of its column vectors. The basis for the column space is the set of the linearly independent columns of the matrix. Elementary row operations preserve the linear relationships between the columns of a matrix. The pivot columns in the RREF of the matrix A correspond to a basis for the column space of the RREF of the matrix A. The pivot columns of the RREF of the matrix A form the basis of the column space of the RREF of the matrix A. The corresponding columns of the matrix A form a basis for the column space of the matrix A. The rank of the matrix A is equal to the columns with leading 1s in the RREF of the matrix A.