I need help with the following questions.
1) show that X & Y are orthogonal in Rn if and only if ||x+y|| = ||x-y||
2)let A be an n x n matrix, find a matrix A for which col A = null A
3) if A is m x n matrix and B is n x m matrix, show that AB = 0 if and only if col B is contained in null A
1) if x & y are orthogonal then x * y = 0 then what?
3) let y be a vector in of B [y1...yn]T then BX = 0 for all X in Rn
col B {y | y= BX}
Y= BX
AY=ABX
but AB = 0
AY = 0X
then wat?
This can only be true if is even. Because by the rank-nullity theorem the rank of a matrix plus its nullity must be n. However, if the column space and nullspace are equal then it means the rank and nullity are the same. And so we have an even number for n.
Let be the following matrix: everywhere, except with for and .
For example, if then the matrix is:
Notice that . As a consequence we see that that the coloumn space is contained in the nullspace. To see this let be in the coloumn space. Then for some . Therefore, . Therefore, is in the nullspace. To show that the coloumn space is equal to the null space it sufficies to show that they have the same dimension. The null space has dimension , that is easy to see. Because the matrix is in row-reduced echelon form and so there are "free" variables that come up in solving the homogenous equation. Immediately we know that that coloumn space has dimension and so this matrix is one such example that you are looking for.
Part of this problem was already addressed in problem #2.3) if A is m x n matrix and B is n x m matrix, show that AB = 0 if and only if col B is contained in null A