Linear Algebra Answers

Consider the relation ~ in R^2 given by (x1,X2)~ (y1,y2) if x1x2~y1y2=1. Check if ~ is an equivalence relation

There are at least three unitary matrices of order 2. True or false with full explanation

Given any n∈N , it is possible to define a linear transformation whose kernel has dimension n.

True or false with full explanation

The function < , > : R^2× R^2→ R : <(x1), (y1)>

<(X2),(y2)>= x1^2+x2^2 define inner product over R^2

True or false with full explanation

If V={(x,y)∈R^2 | x=y} , then (1,0) +V and (0,1)+V are two distinct element of R^2/V.

True or false with full explanation

Check whether or not the set of all symmetric matrix in Mn(R) form a real vector space with respect to the usual addition and scalar multiplication for Mn(R)

Let T: R^2→R^2 be a linear opeator with matrix

[ 7 1]

[-1 1] (w.r.t. the standard basis). Use Cayley haMilton theorem to check whether T is invertible or not. If T is invertible, obtain T^-1(x,y) for (x,y)∈ R^2. If T is not invertible, obtain the minimal polynomial of T.

Is {sinx, cosx, sin(x+π/4)} a linearly independent set over R ? Justify our answer

Apply the fundamental theorem of homomorphism to prove that

R^4/R^2 is isomorphic to R^2

Applications of linear algebra in computer science