Linear Algebra

Find a basis for R(A)^⊥, where R(A) denotes the row space of the matrix
A where A is a 3x4 matrix (1 0 4 0 \ 0 1 2 -1 \ 1 -1 2 1)

Linear Algebra

Find the dual basis for the basis {1,1+x,x²-1} of the vector space P3={a0+a1x+a2x²:a0,a1,a2 belongs to R}

Linear Algebra

Show that if S and T are linear
transformations on a finite dimensional
vector space, then rank (ST)<= rank (T). Also
give examples of linear transformations S
and T for which rank (ST) <rank (T).

Linear Algebra

Find the range space and the kernel of the
linear transformation :
T: R4 --> R4, T(x1, x2, x3, x4) =
(x1+ x2+ x3+ x4, x1+ x2, x3+ x4, 0)

Linear Algebra

Let T : R3-> R3be the linear operator
defined by T(x1, x2, x3) = (x1, x3, -2x2- x3).
Let f(x) = - x³+ 2. Find the operator f(T).

Linear Algebra

1.Let B = (a1,a2, a3) be an ordered basis of
R3 with al= (1, 0, -1), a2= (1, 1, 1),
a3= (1, 0, 0). Write the vector v = (a, b, c) as
a linear combination of the basis vectors
from B.
2.Suppose al= (1, 0, 1), a2= (0, 1, -2) and
a3 = (-1, -1, 0) are vectors in R3and
f : R3 -> R is a linear functional such that
f(al) = 1, f(a2) = -1 and f(a3) = 3. If
a = (a, b, c) E R3, find f(a).

Linear Algebra

Given the basis {(1, - 1, 3), (0, 1, - 1),
(0, 3, - 2)} of R3, determine its dual basis.

Linear Algebra

A is a 3x4 matrix where A= (1 1 0 0 \ -1 3 0 1 \ -3 1 -2 1)
Find an orthonormal basis for the row space of the matrix using the Gram-Schmidt Process

Linear Algebra

Let λ ∈ R be an eigenvalue of an orthogonal matrix A. Show that λ = ±1.
(Hint: consider the norm of Av, where v is an eigenvector of A associated with the
eigenvalue λ.)
Also, find diagonal orthogonal matrices B, C such that 1 is an eigenvalue of B
and −1 is an eigenvalue of C.

Linear Algebra

Show that f(1, 1, 1), (1, – 1, 0), (1, 1, 0)1 is a
basis of R3 . Find a dual basis to this basis.