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 possible linear combinations of the columns of A. It represents the span of the columns of A. In other words, Col A consists of all the vectors that can be obtained by multiplying the columns of A by any scalar and adding them.
On the other hand, the equation Ax = b represents a system of linear equations, where A is the coefficient matrix, x is the variable vector, and b is the constant vector. The solutions of this system of equations are the values of x that satisfy the equation.
The column space of A and the solutions of Ax = b are related, but they are not the same. The solutions of Ax = b are specific values of x that make the equation true, while the column space of A is a set of vectors.
In summary, the column space of A (Col A) is the set of all linear combinations of the columns of A, while the solutions of Ax = b represents specific values of x that satisfy the equation. Therefore, the statement is false.
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