Linear Algebra Review

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Review 1 1 1 − 1 1. For the matrix  1 1  3 2 2 1 6  1 2  5 10 1 a) Find a basis for the null space. b) Find a basis for the column space. 2. If P2 is the vector space of polynomials in t with degree 2 or less and T : P2 → R 2  p(0)   p(1) is defined by T ( p ) =  then a) Prove that T is a linear transformation. b) Find a basis of kernel T c) Describe the range of T ? 3. a) Show that  1  1  B =   ,      2 1  is basis of R2 b)   3  2  2 Given another basis C =    ,    of R , write the change of   2  2  coordinates matrix i) ii) from from B C to C to B 1 1 1   4. Let A= 1 − 1 1   1 1 − 1 i) ii) Find the eigenvalues of A. Diagonalize A, if possible. Show all your calculations. Do not just write a calculator/computer output. 5. Find an orthonormal basis for the subspace of R 5 spanned by  5   3   1          − 3 − 1  1    0 ,  2 , − 1   0   1   3           0   1   2   Show all your calculations. Do not just write a calculator/computer output. 6. Use the Grahm Schmidt process to find an orthogonal basis for the subspace of C[−1,1] spanned by {1,1+x, 1 + x − 2x 2 }. Use 1  fgdx as the −1 inner product of f and g in C[−1,1]. 7. Let A be a square nxn matrix and  an eigen value of A. Let W= v  R n :Av = v . Prove that W is a subspace of R n .   8. Let W be a subspace of a vector space V with a specified inner product. If W ⊥ contains the vectors of V that are orthogonal to W, prove that W ⊥ is a subspace of V. 9. Let T : V spaces V V and →W W Prove that and be a one-one linear transformation between the vector S = v1, v2 ,...., vn  be a linearly independent subset of T (v1 ), T (v2 ),...., T (vn ) is a linearly independent subset of W i) Write the quadratic form 5 x12 − 4 x1x2 + 5 x2 2 in terms of matrices. ii)Find a coordinate transformation so that equation 5 x12 − 4 x1x2 + 5 x2 2 −21=0 transforms into an equation without the product term. 10.
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