Question

QUESTION 2 20 points Save Answer (a) Let A- 101 112 and let T: R 225) T: P = R o via maria menina dentar, TV6 – AR.20 - ( +R be the matrix mapping defined by T(x) = ist wens meer under T is the vector b. and determine whether X is unique (b) Let : R2 + R be the linear transformation that maps the vector - Cinto (6and maps v = ()ino (9) Use the fact that is linear to find S(-3). S(2v) and SE-3u + 2v). 10 let F be the mapping from R to R defined by F(x1, x2) = (3x1 - x2. X1 + 2x2, -5x, +7x2). Show that is linear by finding a matrix that implements the mapping Attach File Browse y como

Answer 1

3 (2) from the that we have to question find it is solution clear of - Ax=b where A = 0

a the system is, de con 2 .A2 2 25/1 + x3 = -3 x + x2 +20₂ = 1 2.0, +2x 1502=3

Let us apply now operations on Bi R2-R, l o !-3 { → R₃ -2R, 10239) 1 R₂ R₃-2R, 32K ii o 1 -31 Ã I Rz 3 RR - R3 (10D 0 1 07. R2 R2 3 0 0 1

rant (A) = ranu () = 3, and therefore system An = b is consistent and can be expressed ao x = -4 X, 2-4 SX 4- 2 2 57 / - 4+0+ - 4+3 + w = (-3) is image of a vector Under such T is that the b.

No del A. - din ad = 5-4 + 2-2 - 1 Fo =) At exists. => Ax=b has unique solution o r sulation a= i. e, the required са 9 we . b) s(-34) = -3 Su ( S is linear transformation) =-35($) S (20) = 25re (:s is linear) =25(8) = 2 (ů) - () s(-34+20) = 5(-34) + S(220) [:s is linear (16):6.52 - 18

o F: R² → IR3 in a mapping defined by F(X1, 2) = (3*, -42, 3, +2X2, -52, +742), (My, XZ) ER. {(1,0), (0,1)) is standard basis of R² {(1,0,0), (0,1,0), (0,0,1)) is standard basis of R? F(1,0) = ( 3, 1,-5) = 3 (1,0,0)+1(0,1,0) +65)(201) F(0,1)=(-1, 2,5) =(-1)(1,0,0) +2 (0,1,0) + 7 (0,0,1) tF] = 13 1-5 7 ) 38,- -5 7 1-5x, +7K₂) This [f]x is the matrix corresponding to mapping F R R ', fince matrix is a linear transformation, F: R² R² is a linear mapping.