FINAL EXAM (RIBET)PEYAM RYAN TABRIZIAN(1) TRUE/FALSE(a) If A is a square invertible matrix, then A and A−1have the same rank(b) If A is an m × n matrix and if b is in Rm, there is a unique x ∈ Rnfor whichkAx − bk is smallest.(c) If A is an n × n matrix, and if v and w satisfy Av = 2v, Aw = 3w, thev × w = 0(d) If the dimensions of the null spaces of a matrix and its transpose are equal,then the matrix is qsquare(e) If A is a 2 × 2 matrix, then −1 cannot be an eigenvalue of A2.(f) I likes the linear algebra portion of this course more than the differential equa-tions portion(g) If 4 linearly independent vectors lie in Span {w1, · · · , wn}, then n must beat least 4.(h) If B is invertible, then the column spaces of A and AB are equal.(i) If A is a matrix, then the row spaces of A and ATA are equal(j) If 2 symmetric n × n matrices A and B have the same eigenvalues, thenA = B(k) If the characteristic polynomial of A is p(λ) = (λ − 1)(λ + 1)(λ − 3)2, thenA has to be diagonalizableDate: Wednesday, December 7th, 2011.12 PEYAM RYAN TABRIZIAN(2) Consider the following vectors:v1=01010, v2=01100, v3=01011Find w1, w2, w3such that {w1, w2, w3} is an orthogonal basis for Span {v1, v2, v3}.(3) Solve the following system of differential equations:(x01(t) = − 2x1(t) + 2x2(t)x02(t) =2x1(t) + x2(t)and x1(0) = −1, x2(0) = 3.(4) Find bases for Nul(A), Row(A), Col(A), where:A =1 1 3 23 1 1 04 2 4 2(5) Find the first 4 terms A0, A1, A2, A3of the Fourier cosine series of f(x) =|sin(x)|Hint: sin(A) cos(B) =12[sin(A + B) + sin(A − B)](6) Solve the following PDE:∂u∂t= 25∂2u∂x20 < x < π, t > 0u(0, t) = u(π, t) = 0 t > 0u(x, 0) = sin(3x) − sin(4x) 0 < x < π(7) Suppose v1, · · · , vnare vectors in Rnand that A is an n × n matrix.If Av1, · · · , Avnform a basis for Rn, show that v1, · · · vnform a basis of Rnand that A is invertible.FINAL EXAM (RIBET) 3(8) Let v1=05−2, v2=123, v3=987Suppose A is the 3 × 3 matrix for which Av1= v1, Av2= 0, Av3= 5v3.Find an invertible matrix P and a diagonalizable matrix D such that A = P
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