P4 is the vector space of polynomials with degree at most four. The basis for the span
of {x2 + x4 , 2x2 + 3x4 , −x2 −3x4} is
Which one of the following three maps is a linear transformation
T1 is a linear transformation.
T2 is a linear transformation.
T3 is a linear transformation.
None of T1, T2 , and T3 is a linear transformation.
Both T1 and T3 are linear transformation.
Ax = b denotes a system of linear equations of
2x1 + x2 = 1 , x1 + 2x2 = 2 , x1 + x2 = 1 . The orthogonal projection of b on the
column space CS(A) of A is
Find the least squares solution xLS of the following
System 2x1 + x2 = 1, x1 + 2x2 = 2 , x1 + x2 = 2
For solving a linear system of differential equations, we can express differential
equation with a matrix form X´ = AX , where A is a n× n matrix and X is a
n×1. If we need to solve a system to have
(1) Please find the solution of X(t) from eAt . Here we define dX (t) dt = X´(t) .
Dirichlet Problem: A particular solution of Laplace’s equation in a bounded plane
region R is determined by appropriate boundary conditions. To find the steady state
temperature in a plate with assigned boundary values, we have to solve a boundary
value problem Uxx +Uyy = 0 (within R ); where
Solve the initial value problem
y1´´= 2y1 + y2 + y1´+ y2´
y2´´= −5y1 + 2y2 + 5y1´− y2´
Here, y1´and y2´ is defined as y1´ = dy1/dt and y2´ = dy2/dt , respectively.
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