Answer to Question #324610 in Abstract Algebra for Jlove

Question

3. Let L: R₂ → R₂ be the linear transformation defined by L ([u¹, u₂]) = [u1, 0]

a. Is [0, 2] in Ker L?

b. Is [2,2] in Ker L?

C. Is [3,0] in range L?

d. Is [ 3,2] in range L?

e. Find Ker L.

£. Find Range L.

Answer 1

$KerL={v∈ℝ²: L(v)=(0,0)}$

(a) $L(0,2)=(0,0)$

Hence, $(0,2)∈KerL$

(b) $L(2,2)=(2,0)≠(0,0)$

Hence, $(2,2)∉KerL$

$RngL={w∈ℝ²: L(v)=W, v∈ℝ²}$

(c) $L(3,b)=(3,0)$ for any $b∈ℝ$

Hence, $(3,0)∈RngL$

(d) There exist no $(a,b)∈ℝ²$ such that $L(a,b)=(3,2)$

Hence, $(3,2)∉RngL$

(e) $L(a,b)=(a,0)=(0,0)$

$=> a=0$

Hence, $KerL={(0,b)∈ℝ²: b∈ℝ}$

(f) $L(a,b)=(a,0)=(x,y)$

$=> x=a, y=0$

$RngL={(a,0)∈ℝ²: y∈ℝ}$