An Introduction To Linear Algebra by Kenneth Kuttler

By Kenneth Kuttler

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0 where the 1 is in the ith position and there are zeros everywhere else. Thus T ei = (0, · · · , 0, 1, 0, · · · , 0) . Of course the ei for a particular value of i in Fn would be different than the ei for that same value of i in Fm for m = n. One of them is longer than the other. However, which one is meant will be determined by the context in which they occur. These vectors have a significant property. 3 Let v ∈ Fn . Thus v is a list of numbers arranged vertically, v1 , · · · , vn . Then eTi v = vi .

If span {v1 , v2 } = V, then there exists v3 ∈ / span {v1 , v2 } and {v1 , v2 , v3 } is a larger linearly independent set of vectors. Continuing this way, the process must stop before n + 1 steps because if not, it would be possible to obtain n + 1 linearly independent vectors contrary to the exchange theorem. This proves the theorem. In words the following corollary states that any linearly independent set of vectors can be enlarged to form a basis. 12 Let V be a subspace of Fn and let {v1 , · · · , vr } be a linearly independent set of vectors in V .

The above discussion stated for general matrices is given in the following definition. 1 Let A = (aij ) and B = (bij ) be two m × n matrices. Then A + B = C where C = (cij ) for cij = aij + bij . Also if x is a scalar, xA = (cij ) where cij = xaij . The number Aij will typically refer to the ij th entry of the matrix, A. The zero matrix, denoted by 0 will be the matrix consisting of all zeros. Do not be upset by the use of the subscripts, ij. The expression cij = aij + bij is just saying that you add corresponding entries to get the result of summing two matrices as discussed above.

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