Let A be a m n matrix Unless stated otherwise all matrices

Let A be a m × n matrix. Unless stated otherwise, all matrices are assumed to have real entries:

a) If some columns of A becomes pivotal columns in RREF(A), are these columns of A linearly independent?

b) If v, w are vectors such that 3v and v + w are linearly dependent, what can you say about v and w?

c) If m < n, that is if Ax = b has fewer equations than unknowns, why is Ax = 0have infinitely many solutions, and what can you say about the number of solutions of Ax = b?

Solution

a) the \"pivot columns\" are the columns that contain the leading 1\'s of the rows.

pivot columns are important, because they form a basis for the column space, which has dimension = rank(A). the number of pivot columns in an mxn matrix is always equal to the number of non-zero rows in a row-reduced matrix.

so they are linearly independent

b) if 3v and (v+w) are  linearly dependent

so 3v= a(v+w) , where a is constant

(3-a) v= a w

or v = (a)/(3-a)*w

then v and w are also  linearly dependent.

c)If m < n, that is if Ax = b has fewer equations than unknowns, why is Ax = 0 have infinitely many solutions,

this is because

it has fewer equations than unknowns

there will be free variable and leading variables.

so we can put different values of free variables and can get different solutions.

what can you say about the number of solutions of Ax = b?

it can have either no solutions or infinite solutions,