lecture 14, August 15, 2018.pdf

# As promised we now prove the following theorem every

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As promised, we now prove the following: Theorem (every matrix has a reduced row echelon form) Let A be an m × n matrix. Then there exists a sequence of row operations that convert A into a matrix R in reduced row echelon form. Proof. ˜ A = 1 a 0 B Here 0 is an ( m - 1) × 1 column vector consisting of all zeros, a is some 1 × n row vector and B , is some ( m - 1) × n matrix. Note that in the case m = 1, the matrices 0 and B are unecessary and ˜ A = 1 a is already is reduced row echelon form.

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Existence of reduced row echelon form As promised, we now prove the following: Theorem (every matrix has a reduced row echelon form) Let A be an m × n matrix. Then there exists a sequence of row operations that convert A into a matrix R in reduced row echelon form. Proof. For m > 1, by the induction hypothesis, B has a reduced row echelon form, that is, there exists a sequence E 1 , . . . , E N of ( m - 1) × ( m - 1) elementary matrices such that E N · · · E 1 B = R , where R is some ( m - 1) × ( m - 1) matrix in reduced row echelon form.
Existence of reduced row echelon form As promised, we now prove the following: Theorem (every matrix has a reduced row echelon form) Let A be an m × n matrix. Then there exists a sequence of row operations that convert A into a matrix R in reduced row echelon form. Proof. Now, the key observation we need must make is that for each elementary matrix E i , 1 i N in this sequence, there exists a larger m × m elementary matrix E 0 i so that E 0 i ˜ A = 1 a 0 E i B = E 0 N · · · E 0 1 ˜ A = 1 a 0 E N · · · E 1 B = 1 a 0 R .

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Existence of reduced row echelon form As promised, we now prove the following: Theorem (every matrix has a reduced row echelon form) Let A be an m × n matrix. Then there exists a sequence of row operations that convert A into a matrix R in reduced row echelon form. Proof. 1 a 0 R Finally, we just using the leading 1’s of R , starting from the left and working to right, to zero out the appropriate entires of a as required by the definition of the reduced row echelon form.
Practice problems 1. Evaluate the following matrix products: d 1 0 0 0 d 2 0 0 0 d 3 a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 , a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a 42 a 43 d 1 0 0 0 d 2 0 0 0 d 3 . 2. A square matrix B is skew symmetric if B T = - B . Let A be a square matrix. Prove that 1 2 ( A + A T ) is symmetric and prove that 1 2 ( A - A T ) is skew symmetric. What is 1 2 ( A + A T ) + 1 2 ( A - A T )?
• Three '08
• ME
• Diagonal matrix, Aik Bkj

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