If A E Mmxn (R), A'A and AA¹ have same positive eigenvalues. 22. If A E Mmxn (R) and B = Mnxm (R), then rank(A) = rank(AB). 23. If A E Mmxn (R), then rank(A) = rank(AA) = rank(AA). 24. If x, y, z E F" and a, c, d E F, then (cx + y, az) = ca(x,z) + a(y, z) 25. T, U:VW are linear, if T(v) = U(v) VvEV, then T = U. 26. T, U:VW are linear, and ß = (v₁, V₂,..., vn) is a basis for V, if T(v₁)= U(v) Vi=1,2,..., n, then T = U. 27. T, U:V V are linear operators. If (T(x), y) = (U(x), y) V x,y € → 4 V, then T = U. 28. Vis inner product space, y, z EV, if (x, y) = (x,z), Vx EV, then y=z. 29. T:V V is linear operators. (V₁, V₂, V3) is an basis for V, if x = a₁v₁ + a2v2 + a3v3, then a₁ = (x, v₁), i = 1,2,3. 30. T:VV is linear operators. If (v₁, V₂, V3) is an basis for V and span(v₁,v₂) = N(T), then N(T) = span(v3).