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