Let L. 1'2 ....... 1'2 be the linear operator dellned hy L (at! + bt + c) = (" _ at!. Using the matri x representing L with respect to the basis [12 + I. t. I) for P" find the eigenvalues and associated eigenvectors of L.
1.3 Vector Equations Vector – ordered lit of numbers − A matrix with only one column R – real numbers that appear as entries in the vectors Vectors in R 2 − R – (read “r-two) exponent 2 denotes that each vector contains two entries; the set of all 2ectors with two entries − two vectors in R are equal if and only if their corresponding entries are equal 4 7 − ≠ 7 4 2 − vectors in R are ordered pairs of real numbers Sum of Vectors (vector addition) 2 − given two vectors u and v in R , their sum in the vector u+v obtained by adding corresponding entries of u and v 2 1 + 2 3 − 1 + = = −2 5 −2 + 5 3 Scalar multiple − given a vector u and a real number c, the scalar multiple of u by c is cu obtained by multiplying each entry in u by c 3 3 15 − = −1 , = 5,then = 5−1 = −5 − scalar – a (real) nu2ber used to multiple a vector or matric Geometric Description of R a − geometric point (a,b) = column vectorb Parallelogram Rule for Addition 2 − if u and v in R are represented as points in the plane, then u+v corresponds to the forth vertex of the parallelogram whose other vertices are u, 0, v 3 Vector in R − 3x1 column matrices with three entries − represented geometrically by points in 3D coordinate space, with arrows from the o
