Linear Algebra Answers

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.

A company that manufactures cars used three different types of steels S1, S2, and S3 for producing three cars C1, C2, and C3. The tons of steel requirement in each type of car is given in below table Cars Steel C1 C2 C3 S1 1 1 1 S2 1 2 4 S3 4 2 3 Determine the number of cars of each type which can be produced using 45, 120 and 130 tones of steel of the three types respectively. (Hint: use gaussian method)