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Worksheet 23 (April 2) DIS 119/120 GSI Xiaohan Yan 1 Review METHODS AND IDEAS Theorem 1. The row rank is equal to the column rank for any matrix. In fact we have dim RowpAq` dim NulpAq“ n and dim ColpAq` dim NulpAq“ n, where the former comes from the orthogonality, and the latter comes from the Rank-Nullity theorem. Remark 1. RowpAq“ NulpAq K , ColpAq“ NulpA T q K . Remark 2. Another way to see the theorem is by row reduction. In fact, both the row rank and the column rank are preserved by row reductions. (Why?) So we may reduce the theorem to the case of RREF. But in RREF both ranks are equal to the number of pivots. Theorem 2. Otrhgonal matrices preserve the inner product. In other words, given an orthogonal matrix U , we have U x ¨ U y “ x ¨ y, @x, y P R n . Remark 3. In particular, this gives ||U x|| “ ||x|| if we take x “ y. 2 Problems Example 1. True or false. ( ) Let U be an orthogonal matrix, then detpU q“ 1. ( ) Let U be an orthogonal matrix and x a vector such that U x and x are linearly dependent, then U x “˘x. ( ) If U is diagonal and orthogonal, then U must be an identity matrix. 1 dim Rowla dimColas pivot free van c Rhu c 1pm g consider AT Row CAT NullAT Colin for A mxn another idea of Nff norms angles in ma 4 745 1455145 Tutu g n I'T.y i x.jo detail Liu In detail.at ug detiuTug z sdetllD t 1 Examine 4 11 detiusa F 4 1 detiut 1 Eigenvalues of I o then ux o soUI 5 orthogonal matrix I'ti u isa multiple of can only be F is ane vector of us 1 or 1 Counterexample 4 1 1 11411 11 1 UI's15
