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Let T: R^3→R^2 be given by:
T(x2,x2,x3)= (X1+x2+x3,x2+x3)
Prove that T is a linear transformation. Also find the rank and nullity of T.
Check whether or not the matrix
A=[1 2 3]
[0 2 3]
[0 0 3]
Is diagonalisable. If it is find a matrix P so that P^-1AP is a diagonal matrix. If A is not diagonalisable, obtain its minimal polynomial.
Obtain a solution set for the linear system
x1-2x2-3x3=0
-2x1+4x2+6x3=0
X1+2x2-5=0
Let T and U be the linear operator on R^2 defined by
T(x1,x2)=(x2,x1) and U(x1,X2)=(x1,0)
A) How would you describe T and U in geometrically
B) give rules like the ones defining..
Does the basis
B = {(1, 0, 1), (1, 0, — 1), (0, 3, 4)}
form an orthonormal basis of R3 with
respect to the standard inner product of
R3 ? Justify your answer. If it doesn't form
an orthonormal basis for R3, apply
Gram-Schmidt process to obtain an
orthonormal basis R3 with respect to the
standard inner product on R3.
Let A and B be n × n matrices with A invertible. Prove that AB and BA have the
same eigenvalues.
Let V be a finite dimensional inner product space over the field C of complex
numbers. Suppose B = {x1, x2, . . . , xn} is an orthonormal basis of V .
Prove that for all x, y ∈ V ,
(x|y) = Xn
i=1
(x|xi)(y|xi).
Let V be the vector space of real 2 × 2 matrices with inner product
(A|B) = tr(B
tA).
Let U be the subspace of V consisting of the symmetric matrices. Find an orthog-
onal basis for U
⊥ where U
⊥ = {A ∈ V | (A|B) = 0 ∀B ∈ U}.
Find all solutions to the following system of linear equations.
x1 − x2 − x3 = 2
2x1 + x2 + 2x3 = 4
x1 − 4x2 − 5x3 = 2
𝑇: ℝ3→ℝ3 defined by 𝑇(𝑥1,𝑥2,𝑥3)=(𝑥1+𝑥2,𝑥2+𝑥3,𝑥3+𝑥1).
