Exercise 10 0 12 B level Prove that 6 o6 0 implies that im 60 C ker 8 Our Goal is to find bases Bim a C Bim 0 which will be helpful in some data analysis later on To start I will finally show you how to compute the kernel more efficiently We will unfortunately be working on a silly example but I ll have you practice this method after you see it So we re going to take 5 augment it vertically with the identity and then column reduce it to an upper triangular matrix 1 1 1 00 1 0 000 0 1 000 0 0 100 O 01 0 001 0 0 0 E colycola col E cols cola cola Bker 81 2 The key is now that the vectors of Q under the zero columns of 6 Q will actually form a basis for ker 6 So if we call v through vs the five columns of 6 Q the four vectors in our case that form a basis for the kernel are 0 0 y 10 0 0 0 11 1 00 01 0 00 00 1 00 0 0 0 0 1 0 04 0 0 1 00 Q I E E 10 01 U 0