28-DEC

Recall:

1. System of linear equations
2. Coefficient and augmented matrices
3. Gaussian Elimination
4. Elementary row transformations
5. echelon form and Reduced Row Echelon Form
   Please refer to the lecture notes on 23rd Dec.

Determinants in and .

The “det” (determinant) is a function. The input of this function (variable) is an matrix (in our case, all entries are numbers), and the output is a number.

In the case:

In the case:

%7B3%20%5Ctimes%203%7D%20%26%3D%20a%7B11%7Da%7B22%7Da%7B33%7D%20%2B%20a%7B12%7Da%7B23%7Da%7B31%7D%20%2B%20a%7B13%7Da%7B21%7Da%7B32%7D%20%5C%5C%20%26-%20a%7B31%7Da%7B22%7Da%7B13%7D-a%7B32%7Da%7B23%7Da%7B11%7D-a%7B33%7Da%7B21%7Da%7B12%7D%20%5C%5C%20%0A%26%3D%20a%7B11%7Da%7B22%7Da%7B33%7D%20%2B%20a%7B12%7Da%7B23%7Da%7B31%7D%20%2B%20a%7B13%7Da%7B21%7Da%7B32%7D%20%5C%5C%20%0A%26-%20a%7B13%7Da%7B22%7Da%7B31%7D-a%7B11%7Da%7B23%7Da%7B32%7D-a%7B12%7Da%7B21%7Da%7B33%7D%5Cend%7Baligned%7D#card=math&code=%5CLARGE%20%0A%5Cbegin%7Baligned%7D%5Cdet%20%28a%7Bij%7D%29%7B3%20%5Ctimes%203%7D%20%26%3D%20a%7B11%7Da%7B22%7Da%7B33%7D%20%2B%20a%7B12%7Da%7B23%7Da%7B31%7D%20%2B%20a%7B13%7Da%7B21%7Da%7B32%7D%20%5C%5C%20%26-%20a%7B31%7Da%7B22%7Da%7B13%7D-a%7B32%7Da%7B23%7Da%7B11%7D-a%7B33%7Da%7B21%7Da%7B12%7D%20%5C%5C%20%0A%26%3D%20a%7B11%7Da%7B22%7Da%7B33%7D%20%2B%20a%7B12%7Da%7B23%7Da%7B31%7D%20%2B%20a%7B13%7Da%7B21%7Da%7B32%7D%20%5C%5C%20%0A%26-%20a%7B13%7Da%7B22%7Da%7B31%7D-a%7B11%7Da%7B23%7Da%7B32%7D-a%7B12%7Da%7B21%7Da_%7B33%7D%5Cend%7Baligned%7D&id=BgHuI).

the order/arrangement for the row-indices, are always . Look at the arrangement for column-indices.

with +: 123, 231, 312;

the number of reverse ordering are (respectively)

with : 321, 132, 213.

%20%3D%203%2C%20%5C%3B%20%5Ctau(132)%3D1%2C%20%5C%3B%20%5Ctau(213)%3D1#card=math&code=%5Ctau%28321%29%20%3D%203%2C%20%5C%3B%20%5Ctau%28132%29%3D1%2C%20%5C%3B%20%5Ctau%28213%29%3D1&id=R6xRo).

The key point is: how many reverse-ordering in this full permutation.

A full permutaion in digits, is an arrangement for the numbers .

Example: n=4. 1234, 3421, 3214, 4312.

. 12354, 43251, 35124, etc…

By a reverse ordering (in a full permutation) we mean that the entries in the -th and -th positions (), say and , has the relation that .

%2B8%2B0-4-(-16)-0%3D%20%20-4#card=math&code=%5Cdet%20%5Cbegin%7Bpmatrix%7D%202%20%26%200%20%26%201%20%5C%5C%201%20%26%20-4%20%26%20-1%20%5C%5C%20-1%20%26%208%20%26%203%20%5Cend%7Bpmatrix%7D%20%3D%20%28-24%29%2B8%2B0-4-%28-16%29-0%3D%20%20-4&id=e9iZS).

determinants in case. %7Bn%20%5Ctimes%20n%7D#card=math&code=%28a%7Bij%7D%29_%7Bn%20%5Ctimes%20n%7D&id=UGpB3).

is a signed-sum (algebraic sum) of terms, these terms can be formed as follows.

1. use the normalization on the row-indices such that in all terms, the row-indices are in the natural ordering 1234…n. ().
2. Then, according to the number of reverse-ordering in the column-indices, put a sign before each term. + for even reverse-ordering, - for odd reverse-ordering.

Usually we write it in the following way:

%20%3D%20%5Csum%20(-1)%5E%7B%5Ctau(j1%20j_2%20%5Cldots%20j_n)%7D%20a%7B1j1%7Da%7B2j2%7D%20%5Cldots%20a%7Bn%20jn%7D#card=math&code=%5Cdet%28a%7Bij%7D%29%20%3D%20%5Csum%20%28-1%29%5E%7B%5Ctau%28j1%20j_2%20%5Cldots%20j_n%29%7D%20a%7B1j1%7Da%7B2j2%7D%20%5Cldots%20a%7Bn%20j_n%7D&id=EIPIm), where is a full permutation of .
The summation runs through all full permutation of .
So there are in total factorial terms.

Concluding remark:

1. We can also define the determinant by making normalization on the column-indices, and look at the reverse-ordering in the row-indicies.
   %20%3D%20%5Csum%20(-1)%5E%7B%5Ctau(i1%20i_2%20%5Cldots%20i_n)%7D%20a%7Bi11%7Da%7Bi22%7D%20%5Cldots%20a%7Bin%20n%7D#card=math&code=%5Cdet%28a%7Bij%7D%29%20%3D%20%5Csum%20%28-1%29%5E%7B%5Ctau%28i1%20i_2%20%5Cldots%20i_n%29%7D%20a%7Bi11%7Da%7Bi22%7D%20%5Cldots%20a%7Bi_n%20n%7D&id=C05mA), where is a full permutation of .
2. In other words, the rows and columns in a determinant are “symmetric”.
3. The computation for determinants is by using the elementary row/column transformations to reduce to an upper-/lower-triagular determinants.
   (1) Switching once, then introducing one minus sign.
   (2) Multiplying a constant to each row/column, then we should also multiply this constant to the det.
   (3) the third kind of elementary transformation does not change the determinant.
4. The elementary column transformations are defined similarly as elementary row transformations.
5. An upper (resp. lower) triangular matrix is a square matrix such that if (resp. ), then .
   . .
6. The triangular matrix has determinant equal to the product of all its diagonal entries.

• Comments on homework.

partial fraction decompositon trick to calculate indefinite integrals.

%5E2%7D%20%3D%20%5Cfrac%7B-1%7D%7Bx-3%7D%2BC#card=math&code=%5Cdisplaystyle%20%5Cint%20%5Cfrac%7Bd%20x%7D%7B%28x-3%29%5E2%7D%20%3D%20%5Cfrac%7B-1%7D%7Bx-3%7D%2BC&id=BvrQ3) %7D%2BC#card=math&code=%5Cfrac%7B1%7D%7B%5Cln%28x-3%29%7D%2BC&id=zmYo7) (wrong)

%5E2%7D%20dx#card=math&code=%5Cint_3%5E6%20%5Csqrt%7B9-%28x-3%29%5E2%7D%20dx&id=Ci6vu) The integrand %5E2%7D#card=math&code=%5Csqrt%7B9-%28x-3%29%5E2%7D&id=l2L7o)

The problem of not following the instructions!

%5E2%7D%20dt#card=math&code=%5Cint_0%5E1%20%5Cfrac%7B1-2t%7D%7B1%2B%28t-1%2F2%29%5E2%7D%20dt&id=DwBDJ) because %5E2)%20%3D%202(t-1%2F2)d(t-1%2F2)%20%3D(2t-1)%20dt#card=math&code=d%28%201%2B%28t-1%2F2%29%5E2%29%20%3D%202%28t-1%2F2%29d%28t-1%2F2%29%20%3D%282t-1%29%20dt&id=WTP9K)

let The interval is symmetry about the origin. The intergrand is an odd function. The original integral must vanish.