.oo T-Mobile 3:13 AM Moodle 2. Define a function T : R4 R2 be defined a8 T(zi.r2

Question

.oo T-Mobile 3:13 AM Moodle 2. Define a function T : R4 R2 be defined a8 T(zi.r2)-(a-24-rs). Show that T is a linear transformation. 3. Let T : R5 R3 be defined as T(a, a2, a3, a4, as) = (aut 2a2-a3:-aat 33,-al-a2-3a3) (a) Find a basis for range and kernel of T (b) Find the ty and the rank of T (c) Is T one-to-one? Explain. (d) Is T onto? Explain. (e) Is T an isomorphis Explain 4. Define a function T : R2 R4 be defined as T(a, a2)-(al, 2a1. 3a1. 4a). (a) Show that T is a ar transformation. (b) Find a basis for range and null space of T (c) Find the nulty and the rank of T (d) Is Toet-e? Explain (e) Is T onto? Explain (f) Is T an isomorphism? Explain.

Explanation / Answer

2. T:R4 R2 is defined by T(x1,x2,x3,x4) = (x1-x4, x2-x3). Let X =(x1,x2,x3,x4) and y =(y1,y2,y3,y4) be 2 arbitrary elements of R4 and let be an arbitrary scalar. Then T(X+Y) = T( (x1,x2,x3,x4) +( y1,y2,y3,y4)) = T (x1+y1, x2+y2, x3+y3,x4+y4) = (x1+y1 -x4- y4, x2+y2-x3-y3)= Also T(X)+T(Y)= (x1-x4, x2-x3)+ (y1-y4, y2-y3)= (x1-x4, x2-x3) +(y1-y4, y2-y3)= (x1-x4 +y1-y4, x2-x3+ y2-y3)= (x1+y1 -x4- y4, x2+y2-x3-y3)= T(X+Y). Thus, T preserves both sca;ar multiplication and vector addition. Hence T is a linear transformation.

Please post Q3 again separately.

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