Problem 4: Consider the set of all vectors in R4 which are mutually orthogonal to the vectors <3,4,−1,1> and p4 <1,1,0,2

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Problem 4 Consider The Set Of All Vectors In R4 Which Are Mutually Orthogonal To The Vectors 3 4 1 1 And P4 1 1 0 2 1 (177.13 KiB) Viewed 35 times

Problem 4: Consider the set of all vectors in R4 which are mutually orthogonal to the vectors <3,4,−1,1> and p4 <1,1,0,2 >. (a) The first thing you need to do is determine the form of all vectors in this space. Hints on how to proceed: ● You need vectors <a,b,c,d > with the property that <a,b,c,d > is orthogonal to <3,4,-1,1> and <a,b,c,d > is orthogonal to <1,1,0,2>. There's a vector equation that defines "orthogonal" and this will set up two equations. That means you have two equations with four unknowns - there are infinitely many solutions for a,b,c,d, but you can parameterize. Rref the system, and parameterize so that a and b are expressed in terms of c and d. That means <a,b,c,d > becomes <[thing with c's and d's],[thing with c's and d's],c,d > and you have the form of your space and can proceed as usual with things in this form. (b) Prove that this set is a subspace of R* by explicitly verifying the closure axioms. By this, I mean that by now we have a theorem that says all sets generated in this form are subspaces and satisfy closure, but I do NOT want you to just cite the theorem. Verify from the definitions of closure on vectors in the form of your space. (c) Find a basis for the subspace, and give the dimension of the subspace.