This applet shows the row, column, and solution spaces of a 3x3 matrix M. The row space is shown in blue, the column space is shown in orange, and the solution space is shown in green. [br][br]You can edit the matrix to view how these spaces change, and you can toggle the visibility of the three spaces by clicking the checkboxes on the right.[br][br]Some things you may notice are:[br]- The row and column spaces always have the same dimension, equal to the rank of M[br][br]- The dimension of the row/column spaces and the dimension of the solution space always sums to 3, in accordance with the rank-nullity theorem[br][br]- The row space is always orthogonal to the solution space of M[br][br][br]Here are some interesting matrices to try:[br]⌈ 1 2 -1 ⌉[br]| 2 4 -2 |[br]⌊ 1 2 -1  ⌋[br]Before you put this in the applet, look at the columns of this matrix - what do you notice? Can you guess what the column space will look like?[br][br][br]⌈ 1 2 -4 ⌉[br]| 2 -1 7 |[br]⌊ 1 0 3 ⌋[br]What is the solution space in this case? (Hint: remember that the span of an empty set is the 'trivial' subspace consisting of just the [b]0[/b] vector.)[br][br]The 0 matrix:[br]⌈ 0 0 0 ⌉[br]| 0 0 0 |[br]⌊ 0 0 0 ⌋[br]What is the row and column space in this case?