MAT 208 - Final Exam
homogeneous linear systems - Answer-always consistent
linear system of 3 equations - Answer-only 1 unique solution
trace of square matrix A = [{2,3}, {5,0}] - Answer-2 + 0 = 2
sum of 2 n x n symmetric matrices - Answer-symmetric
if A is any n x n matrix - Answer-A + A^T is symmetric
standard matrix representing linear transformation L -> R^3 defined by L ([{u1},{u2},{u3}]) = [{u1 + 4u2},
{-u^3}, {u2 + u3}] - Answer-L(e1) = L[{1},{0},{0}] = [{1},{0},{0}]
L(e2) = L[{0},{1},{0}] = [{4},{0},{1}]
L(e3) = L[{0},{0},{1}] = [{0},{-1},{1}]
det(AA^T) = - Answer-det (A^2)
if matrix A is orthogonal - Answer-matrix A^2 is orthogonal too
if matrix A is 4 x 4 matrix and c is a scalar - Answer-det(cA) = c^4 det(A)
, if det(A) = 5 - Answer-Ax = 0 has only trivial solution
if B = PAP^-1 and P is nonsingular - Answer-det(B) = det(A)
show u * v = 0 iff ||u+v|| = ||u-v|| - Answer-||u+v||^2 = ||u-v||^2
||u -v|| = (u, u) - 2 (u, v) + (v, v)
u^2 + 2 (u, v) + v^2 = u^2 - 2 (u, v) + v^2
4 (u, v) = 0
(u, v) = 0
u*v=0
for what values of k are vectors t+4 and 2t+k^2-1 linearly independent? - Answer-a1 (t+4) + a2 (2t+k^2-
1) = 0
k ≠ +- sqrt(9)
determine r values so that x = 1, y = -1, z = r
x - 2y + 3z = 3
4x + 5y -z = -1
6x + y + 5x = 5 - Answer-3 + 3r = 3 -> 3r = 0
-1 - r = -1 -> -r = 0
5 + 5r = 5 -> 5r = 0
r=0
homogeneous linear systems - Answer-always consistent
linear system of 3 equations - Answer-only 1 unique solution
trace of square matrix A = [{2,3}, {5,0}] - Answer-2 + 0 = 2
sum of 2 n x n symmetric matrices - Answer-symmetric
if A is any n x n matrix - Answer-A + A^T is symmetric
standard matrix representing linear transformation L -> R^3 defined by L ([{u1},{u2},{u3}]) = [{u1 + 4u2},
{-u^3}, {u2 + u3}] - Answer-L(e1) = L[{1},{0},{0}] = [{1},{0},{0}]
L(e2) = L[{0},{1},{0}] = [{4},{0},{1}]
L(e3) = L[{0},{0},{1}] = [{0},{-1},{1}]
det(AA^T) = - Answer-det (A^2)
if matrix A is orthogonal - Answer-matrix A^2 is orthogonal too
if matrix A is 4 x 4 matrix and c is a scalar - Answer-det(cA) = c^4 det(A)
, if det(A) = 5 - Answer-Ax = 0 has only trivial solution
if B = PAP^-1 and P is nonsingular - Answer-det(B) = det(A)
show u * v = 0 iff ||u+v|| = ||u-v|| - Answer-||u+v||^2 = ||u-v||^2
||u -v|| = (u, u) - 2 (u, v) + (v, v)
u^2 + 2 (u, v) + v^2 = u^2 - 2 (u, v) + v^2
4 (u, v) = 0
(u, v) = 0
u*v=0
for what values of k are vectors t+4 and 2t+k^2-1 linearly independent? - Answer-a1 (t+4) + a2 (2t+k^2-
1) = 0
k ≠ +- sqrt(9)
determine r values so that x = 1, y = -1, z = r
x - 2y + 3z = 3
4x + 5y -z = -1
6x + y + 5x = 5 - Answer-3 + 3r = 3 -> 3r = 0
-1 - r = -1 -> -r = 0
5 + 5r = 5 -> 5r = 0
r=0
