A denotes an m×n matrix. Determine whether the statement is true or false. Justify your answer:The column space of A, Col A, is the set of all solutions of Ax=b.
The statement is false
The statement is false. The column space of a matrix A, denoted as Col A, is the set of all linear combinations of the columns of A. It represents the span of the columns of A.
On the other hand, Ax = b represents a system of linear equations, where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of constants. The solution to this system of equations, if it exists, represents a specific point or vector x that satisfies the equations.
Therefore, the column space of A, Col A, is not the same as the set of all solutions of Ax = b. The column space represents a subspace in the vector space defined by the columns of A, while the set of solutions of Ax = b represents specific points or vectors satisfying the given system of equations.
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