S lis points) True : False. EXPLAIN) A i inveamble and v. 7 V, then AV, AV, 1 IT
ID: 3137651 • Letter: S
Question
S lis points) True : False. EXPLAIN) A i inveamble and v. 7 V, then AV, AV, 1 IT a) The subset A of paly nam tals of degree 2 that are zero at x=0 , x=1 Simultancovsly form a subspace of 2 { dim 4 - 2 c) Let A be a 3x3 matnx w/ rank 2 (2 pivots) { Columns VI, v, 13. TË h is a Vetor such that Ax = 5 in canristant, then Span {, vai Vz, b3 = 123 d) The number of pivots of the matrix Ab is never larger than the number of pivots af A. Ant : points wil be the calumns of A the relation b-w given if you dera be the # of prots - > the relation bw the eglu mng of AB at a matx c 0 a nonzero 2x2 mat is tho rank ofExplanation / Answer
5. (a) True. If A is invertible and Av1 = Av2, then on multiplying to the left by A-1, we get A-1Av1 = A-1Av2 or, v1 = v2.
(b). False. If p(x) = ax2+bx +c is an element of H , then c = 0 and a+b = 0 so that p(x) = ax2-ax = a(x2-x). Thus, any arbitrary element of H is a scalar multiple of x2-x so that dim(H) = 1.
(c ). True. If the rank of A is 2, then only 2 of v1,v2,v3 are linearly independent and dim(span{v1,v2,v3}) = 2. Also, if the equation AX = b is inconsistent, then b ? span { v1,v2,v3} so that dim(span { v1,v2,v3,b}) = 3 and hence span { v1,v2,v3,b} =R3.
(d). True. The columns of AB are linear combinations of the columns of A as each column of AB equals A times the corresponding column of B.
(e). Let A =
a
b
c
d
where a,b,c,d are arbitrary real numbers, not all 0, so that A ?0. Then AA =
a2+bc
ab+bd
ac+cd
d2+bc
and AAA =
a(a2+bc)+c(ab+bd)
b(a2+bc)+d(ab+bd)
a(ac+cd)+c(d2+bc)
b(ac+cd)+d(d2+bc)
It may be observed that id AA ? 0, then a2+bc, ab+bd,ac+cd, d2+bc cannot all be 0. Thus, if AAA = 0, then a,b,c,d are all 0 so that A = 0, which is a contrasiction. Hence, we cannot have a 2x2 non-zero matrix A such that A2 ? 0, but A3 = 0
a
b
c
d
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