Suppose A is a 5 times 4 matrix such that the only solution

Suppose A is a 5 times 4 matrix such that the only solution to AX = 0 is the trivial solution. Are the columns of linearly independent? Do the columns of A span IR^s? B Let T:IR^4 rightarrow IR^3 be a linear transformation. What is the size of the matrix representation of T? Suppose the matrix representation of T has 3 point positions. Is T a one-to-one transformation? Does T map IR^4 onto R^3? C Let {u, v, w} be a linearly independent subset of IR^4 is {u, v, w, o} also a linearly independent subset of IR^4?

Solution

A) if the matrix A has only the trivial solution to Ax=0 so the columns of A are linearly independant. the columns cannot span R5 because you would need 5 linearly independant columns to span in R5

B) the size of the matrix will be 3x4 because of the theorems of linear transformations. Since A has more columns than rows, so T is not one to one. Since A has a pivot in each row, the rows of A maps R4 onto R3

C)No, it will not be linearly independant, since some columns will not contain leading entries, then the system has nontrivial solutions and will be linearly dependant