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- Find a basis for R2 that includes the vector (2,2).4. Find a subset of the vectors v1= ( 0, 2 2, 4), v2= ( 1,0,-1,-3), v3= ( 2,3,1,1 ) and v4= ( -2, 1, 3, 2 ) that forms a basis for the space spanned by these vectors. Explain clearly.Show that {u1, u2, u3} is an orthogonal basis for ℝ3. Then express x as a linear combination of the u's. NOTE: for the part where it says "Since each ____ is ___, the vectors _____ From the theorem above, this proved that the vectors are also _____ THE OPTIONS FOR the first blank space with the little down arrow is "difference" or "inner product" THE OPTIONS FOR the second blank space with the little down arrow is "form an orthogonal set", "all have lenght one", "are uniformaly spaced", or form a basis THE OPTIONS FOR the third blank space with the little down arrow is "of lenght one", "a basis", "evenly spaced", "an orthogonal set"
- Show that B = {v1=(1,0,0), v2=(3,7,-2), v3=(0,4,1)} is a basis for R3Let S ={v1 , v2 ,...,vk } be an orthonormal basis for the Euclidean space V and {a1 ,a2 ,...,ak } be any set of scalars none of which is zero. Prove thatT ={a1 v1 ,a2 v2 ,...,ak vk }is an orthogonal basis for V . How should the scalars a1,a2 ,...,ak be chosen so that T is an orthonormal basis for V ?Verify that the set in Example 6 is a subspace. Find a basis for thissubspace. Is {x2 + x + 1, x + 5, 3} a basis?
- Determine whether (1,1,1,1),(1,2,3,2),(2,5,6,4),(2,6,8,5) form a basis of R^4 (R). If not, find the dimension of the subspace they span.Reduce the linear operator matrix to Jordan form. Build canonicalbasis. To control the correctness of the construction of the canonical basis, use the relation P A′ = AP, where A is the given matrix, A′ is the Jordan form of the matrix, P istransition matrix to the canonical basis.Find the basis for the orthogonal complement V: v1=(1,-3,3,5) v2=(2,-5,9,3)