Can {u+v, a*u} Form a Basis for Vector Space V?

Thread starter loli12
Start date Sep 14, 2005
Tags

Basis

AI Thread Summary

To determine if {u+v, a*u} forms a basis for vector space V, it is essential to establish that these vectors are linearly independent and span V. Given that {u, v} is already a basis, u and v are linearly independent and span V. The linear independence of {u+v, a*u} can be shown by demonstrating that no scalar multiples can express one vector as a combination of the other. Additionally, since u and v span V, the combination of u+v and a*u will also span V. Thus, {u+v, a*u} indeed forms a basis for V.

Sep 14, 2005

loli12

I have to show that for 2 distinct vectors u and v of a vector space V, for which {u, v} is a basis for V and a and b are nonzero scalars, then {u+v, a*u} is also basis for V.

Please help!

Last edited by a moderator: Sep 14, 2005

Physics news on Phys.org

Sep 15, 2005

quasar987

Science Advisor

Homework Helper

Gold Member

From memory (and common sense), u and v are a basis if they are linearly indep. and they span V right?

So you need to show that provided that u & v satisfy these two conditions, then (\Rightarrow) u+v & a*u also satisfy them.

Thread 'I've come across another fluid pressure problem I don't understand'

Neglect gravity other than that of water; neglect friction. F1=ρgsh=1000*10*1*(1+4)=50000 N; F1=ρgsh=1000*10*0.1*4=4000 N; F3=50000-4000=46000 N?

View full post »

Thread 'Struggling to understand the concept of this question (ice cube in water optics question)'

TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...

View full post »

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...

View full post »