i've been having some trouble with my linear algebra homework and im wondering if you guys could give me some insight or tips on these problems:(adsbygoogle = window.adsbygoogle || []).push({});

Let v be any vector from V, and let a be any real number such that av=0. Show that either a=0 or v=0.

- i was thinking about assuming the hypothesis that av=0, and then proving the implication by showing that a=0 and v=0 are valid solutions, and then that no other distinct solutions exist. but i ran into trouble with proving the "uniqueness" of the two solutions. ****any other methods or approaches would be great to hear

Similar to the first one, if av=v, show that a=1 or v=0. kinda stuck on this one as well.

Let P(R) be the vector space of polynomials in z of degree at most 2 with real coefficients. Thus P(R) = {a + bz + cz^2: a,b,c are all real numbers}

- give an example of a subset U of P(R) that is closed under scalar multiplication but is not a subspace.

^ ok this one is giving me a problem. if scalar multiplication is closed, this means additive inverses exist. this means that the zero vector is also in this subset. i

- give an example of a subspace U of P(R) that is proper, ie not empty and not the entire space

- find another subspace W such that U(direct sum)W = P(R)

Suppose U1, U2, U3 are subspaces such that V= U1+U2+U3, formulate a condition in terms of suitable intersections of U1, U2 and U3 such that V = U1(Direct Sum)U2(direct sum)U3. and then generalize for k subspaces.

to me, it's pretty tough stuff....

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# Homework Help: Vector Spaces and Subspaces

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