{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

assign5_soln

# assign5_soln - Math 136 Assignment 5 Due Wednesday Feb 24th...

This preview shows pages 1–3. Sign up to view the full content.

Math 136 Assignment 5 Due: Wednesday, Feb 24th 1. Let A = 1 2 - 1 3 - 2 - 1 , B = 1 0 0 0 2 - 3 , C = 5 - 1 2 1 - 1 2 . Determine the following a) 2 A - B Solution: 2 A - B = 2 4 - 2 6 - 4 - 2 - 1 0 0 0 2 - 3 = 1 4 - 2 6 - 6 1 b) A ( B T + C T ) Solution: B T + C T = 1 0 0 2 0 - 3 + 5 1 - 1 - 1 2 2 = 6 1 - 1 1 2 - 1 . So A ( B T + C T ) = 1 2 - 1 3 - 2 - 1 6 1 - 1 1 2 - 1 = 2 4 18 2 c) BA T + CA T Solution: BA T + CA T = ( B + C ) A T = [ A ( B + C ) T ] T = 2 18 4 2 . 2. Prove that if x M (3 , 2) and a, b R are scalars, then ( a + b ) x = a x + b x . Solution: Let x = x 11 x 12 x 21 x 22 x 31 x 32 . Then ( a + b ) x = ( a + b ) x 11 x 12 x 21 x 22 x 31 x 32 = ( a + b ) x 11 ( a + b ) x 12 ( a + b ) x 21 ( a + b ) x 22 ( a + b ) x 31 ( a + b ) x 32 = ax 11 ax 12 ax 21 ax 22 ax 31 ax 32 + bx 11 bx 12 bx 21 bx 22 bx 31 bx 32 = a x + b x. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 3. Determine which of the following mappings are linear. Find the standard matrix of each linear mapping. a) f ( x 1 , x 2 , x 3 ) = ( x 1 + x 2 + 1 , x 3 , 0). Solution: Observe that f [(1 , 0 , 0)+(1 , 0 , 0)] = f (2 , 0 , 0) = (3 , 0 , 0), but f (1 , 0 , 0)+ f (1 , 0 , 0) = (2 , 0 , 0) + (2 , 0 , 0) = (4 , 0 , 0). Thus f is not linear. b) f ( x 1 , x 2 ) = (0 , x 1 + 2 x 2 , x 2 ). Solution: We have f [ c ( x 1 , x 2 ) + ( y 1 , y 2 )] = f ( cx 1 + y 1 , cx 2 + y 2 ) = (0 , ( cx 1 + y 1 ) + 2( cx 2 + y 2 ) , cx 2 + y 2 ) = c (0 , x 1 + 2 x 2 , x 2 ) + (0 , y 1 + 2 y 2 , y 2 ) = cf ( x 1 , x 2 ) + f ( y 1 , y 2 ) . Hence, f is linear. To find the standard matrix we find f (1 , 0) = (0 , 1 , 0) , f (0 , 1) = (0 , 2 , 1) .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

assign5_soln - Math 136 Assignment 5 Due Wednesday Feb 24th...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online