How many pivot columns must a 6 times5 matrix have if its co

How many pivot columns must a 6 times5 matrix have if its columns are linearly independent? Justify your answer. 5. How many pivot columns must a 4 times 6 matrix have if its columns span R4? Justify your answer. 6. Construct a 3times2 matrix A such that Ax = 0 has only the trivial solution. Explain how you did it. Construct a 3 times 2 matrix B such that Bx = 0 has nontrivial solutions. Explain how you did it.

Solution

Question 4)

If columns of a 6x5 matrix are linearly independent, then, the basis of matrix will have number of vectors equal to the number of columns of the matrix. That is, the basis will have 5 vectors. Hence, matrix will have 5 pivoting columns.

Question 5)

We are given that the columns of 4x6 matrix span R⁴. This only means that basis of matrix must have 4 vectors. This means that the given matrix must have 4 pivoting columns.