Question 1 Question 2 find the coordinate vector of x= [ -4,3,2] with respect to

Question

Question 1

Question 2

find the coordinate vector of x= [ -4,3,2] with respect to the basis

Question 3

is a basis of the space of upper- trangular 2x2 matrices.

Explanation / Answer

(1) Let [-2 -3] = a[5 -3] + b[1 3] [-1 -2] = c[5 -3] + d[1 3] => 5a+b = -2 and -3a+3b = -3 Solving we get a = -1/6 , b = -7/6 Also 5c+d = -1 and -3c+3d = -2 Solving we get c = -1/18 , d = -13/18 P = [ a c b d ] = [ -1/6 -1/18 -7/6 -13/18 ] (2 rows written in same line) (2) Let [-4 3 2] = a[1 0 0] + b[8 1 0] + c[4 -7 1] => a+8b+4c = -4 b-7c = 3 c = 2 Solving we get a = -148 , b = 17, c = 2 Hence [x]_B = [ -148 17 2 ] (written horizontal instead of vertical) (3) Let [ 1 9 0 -9 ] = a [ -1 -1 0 0 ] + b [ 0 -1 0 2 ] + c [ 0 0 0 2 ] (2 rows written in same row) => -a = 1 -a-b = 9 0 = 0 -2b+2c = -9 Solving we get a = -1 , b = -8 , c = -25/2 Hence {M}_B = [ -1 -8 -25/2 ] (written horizontal instead of vertical)

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