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[FONT=MathJax_Math]how we get, N[FONT=MathJax_Math]ρ[FONT=MathJax_Main]≤[FONT=MathJax_Main]1

we have to prove. please explain clearly.

A=[cos^2(theta)-sin^2(theta), -2sin(theta)cos(theta) ; 2sin(theta)cos(theta), cos^2(theta)-sin^2(theta)] (A is 2*2 matrix.)

Sorry, I'm not getting it. Can you explain in brief.

1.Construct a matrix whose null space consists of all linear combination of the vectors, v1={1;-1;3;2} and v2={2,0,-2,4} (v1,v2 are column vector).2.The equation x1+x2+x3=1 can be viewed as a linear system of one equation in three unknowns. Express its general solution as a particular solution...

Prove that if \(P\) is a stochastic matrix whose entries are all greater than or equal to \(\rho\), then the entries of \(P^{2}\) are greater than or equal to \(\rho\).

1.(a) Give an example of 3*3 matrix whose column space is a plane through the origin in 3-space (b) what kind of geometry object is the null space and row space of your matrix 2. Prove that if a matrix A is not square, then either the row vectors or the column vectors of A are linearly dependent.

how to proof if second statement is true then first statement is false.

Let T:V->V be a linear operator on an n-dimensional vector space. Prove that exactly one of the following statements holds: (i) the equation T(x)=b has a solution for all vectors b in V. (ii) Nullity of T>0

Prove that [FONT=MathJax_Math]F[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main]), [FONT=MathJax_Math]C[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main])...

Prove that \(F({\infty},-{\infty})\), \(C({\infty},-{\infty})\), \(C^{\infty}({\infty},-{\infty})\) and \(C^m({\infty},-{\infty})\) are infinite dimensional.

yes it is \(R^{\infty }\)

1. If multiplication by A rotates a vector X in the xy-plane through an angle (theta). what is the effect of multiplying x by A^T ? Explain Reason.

Prove that \(R^{\infty}\) is infinite dimensional.

what is U(0,1) ? Please explain me, i couldn't understand.