column space and null space and row reductionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=Here are 3 column vectors in space, and they are coplanar but no two are colinear (proportional).PkkiQUc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjL0kkJWlkR0YkIik7XkduRow reduction takes linearly independent combinations of the rows, which preserves the relationships among the columns (same solutions of reduced linear system as of the original linear system).LUk2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRzYiNiNJIkFHRiQ= In the row reduced matrix it is obvious that the third column (the only free column) is twice the first column plus the second column (the 2 leading columns), so a basis of the column space of all possible linear combinations of the 3 column vectors consists of the first two (leading) columns of the original matrix. In fact any two of the 3 vectors would do, but our algorithm works from left to right and hence privileges the first two of the 3.On the other hand, we can take new linearly independent combinations of these vectors to get a basis consisting of new vectors that do not belong to the set, but which span the same column space. This is accomplished by tranposing the matrix to make them rows, and then row reducing, and then the nonzero rows are a basis of the row space of the transposed matrix, and hence when transposed back to column vectors, become a basis of the column space.QyctSSpUcmFuc3Bvc2VHNiI2I0kiQUdGJSIiIi1JNlJlZHVjZWRSb3dFY2hlbG9uRm9ybUdGJTYjSSIlR0YlRigtRiRGKw==The first two columns are a basis of the column space, and that is what the ColumnSpace command gives us: a list of the basis vectors.LUksQ29sdW1uU3BhY2VHNiI2I0kiQUdGJA==As a list of column vectors we can pop them into the Matrix command and get the augmented matrix, which explains why the list output might be more convenient than the set output like NullSpace command below.LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJIiVHRic=Notice that this basis if we ignore the third component, which means projecting the 3-vectors down to the x-y plane, we get the standard basis of the plane. This gives a geometric meaning to the row reduction process. The actual plane is tilted. The row reduction process picks out those two vectors in the tilted plane which lie above or below the standard basis of the horizontal plane.The solution space of the linear system LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyRjUtRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZWRjUtSSNtbkdGJDYkUSIwRidGOUY5 is called the null space or kernel of the matrix. We solve this matrix equation by the pair of commands.Ly1JJDwsPkc2IjYlJkkieEdGJTYjIiIiJkYoNiMiIiMmRig2IyIiJC1JM0JhY2t3YXJkU3Vic3RpdHV0ZUdGJTYjLUk2UmVkdWNlZFJvd0VjaGVsb25Gb3JtR0YlNiMtSSQ8fGdyPkdJKF9zeXNsaWJHRiU2JEkiQUdGJS1JJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0Y5NiRGMCIiIQ==The coefficient vectors of the parameters (only one in this case) form a basis of the solution space, but the output of the NullSpace command are column vectors which are listed as a set "{...}" of vectors rather than a list "[...]" of vectors like the ColumnSpace command. Each such coefficient basis vector describes an independent linear relationship among the original column vectors of the matrix. In this example the last vector minus twice the first vector minus the second vector is the zero vector as we saw above.LUkqTnVsbFNwYWNlRzYiNiNJIkFHRiQ=But we can convert this to a list if we wish, which pops into the Matrix command to make a matrix of these columns augmented together (but there is only one here).QyUtSShjb252ZXJ0RyUqcHJvdGVjdGVkRzYkSSIlRzYiSSVsaXN0R0YlIiIiLUknTWF0cml4RzYkRiVJKF9zeXNsaWJHRig2I0YnThe dimension of the null space is the number of free variables, which in turn is the total number of variables minus the number of leading variables. Thus the sum of the dimensions of the column space and the null space is equal to the total number of variables.If we just use the LinearSolve command instead of the rowreduction approach, we get the same solution but with a different naming convention for the parameters.LUksTGluZWFyU29sdmVHNiI2JEkiQUdGJC1JJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkIiIkIiIhIf we had more than one parameter in the solution, we would see that one uses subscripts which increase from the bottom up, and the other from the top down.TTdSMApJNVJUQUJMRV9TQVZFLzUwNzE1MjkyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKiIkIiQiIiIiIiMiIiRGKEYpRiciIiUiIihGK0Ym