Linear Algebra Answers

In R^4, let U=span((1,1,0,0),(1,1,1,2)). Find u"\\in"U such that ||u-(1,2,3,4)|| is as small as possible.

suppose u is a subspace of r^4 defined by u=span((1,2,3,-4),(-5,4,3,2)). find an orthonormal basis of u and an orthonormal basis of u^⊥

suppose u is a subspace of r^4 defined by u=span((1,2,3,-4),(-5,4,3,2)). find an orthonormal basis of u and an orthonormal basis of u^π

suppose T€ L(R^3) has an upper-triangular matrix with respect to the basis (1,0,0),(1,1,1),(1,1,2). Find an orthonormal basis of R^3 using the usual inner product on R^3 with respect to which T has an upper-triangular matrix.

Suppose T€L(V) and dim range T =k. Prove that T has at most k+1distinct eigenvalues.

suppose u, v € v. prove that ||au+bv||=||bu+av|| for all a, b € r if and only if ||u||=||v||.

Suppose T "\\in" L(V) and dim range T = k. Prove that T has at most k +1 distinct eigenvalues.

Find vectors u,v "\\in" R² such that u is a scalar multiple of (1,3), v is orthogonal to (1,3), and (1,2) = u +v.