Question 2T Circle the correct answers. For this question, working is not requir

Question

Question 2T Circle the correct answers. For this question, working is not required and will not be marked In parts (a)-(c), A E M(Q) for some fixed n E N, and A is non-singular (i.e A-1 exists) (a) Every linear system Ax b has solution x-A-^b.TRUE/ FALSE (b) If Axi = Ax2 then xi = X2. TRUE / FALSE (c) The function f : Qn ? Qn defined by f(x) = Ax is surjective. TRUE / FALSE For parts (d)-(f) recall that for any n, m e N, a function f : Qn-+ Qn, is linear if and only if (d) For any A e Mmxn the function f : ?-+ Qm defined by f(x)- Ax is linear. TRUE FALSE (e) If the function f : Qn 0" is linear then 3A ? MmxnYx e Qn f(x) Ax. TRUE FALSE (f) The function f : Q ? Q defined by f(z) =-r is linear. TRUE / FALSE

Explanation / Answer

Solution:

2.

(a) TRUE.

(b) TRUE

Ax1=Ax2 implies, A-1Ax1= A-1Ax2 which gives, x1= x2

(c) f is surjective. Infact f is bijective.

(d) TRUE

(i) f(x1+x2)= A(x1+x2) = Ax1+Ax2 = f(x1) + f(x2) and

(ii) f(cx) = A(cx) = cAx = cf(x)

(e) TRUE

(f) TRUE

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