Let P² be the vector space of all polynomials with real coefficients of degree at most 2, with the usual operations. a) Show that the set of vectors S = {x² + 1, 3x − 14, -5x² - 9x +37} is linearly dependent. b) Find a basis for P² which includes the first two vectors of S (that is, v₁ = x² + 1 and v₂ = 3x – 14).

Consider the following "standard" basis of R³ over R: --000 = Let q be the real quadratic form (from R³ to R) defined via: q(x1, x2, x3) = 9x² + 7x² + 2x3 + 18x1x2 + 18x1x3 + 10x2x3 Determine [q], the symmetric matrix representing the quadratic form q in terms of the basis &:

The set B = {-(2 + 2x²), - (6+4x+6x²), 18 + 8x + 16x²} is a basis for P₂. Find the coordinates of p(x) = - (32 + 8x +26x²) relative to this basis: [p(x)] B =

The set B = {− (3 + 4x²), − (6 + 3x + 8x²), 8 + 6x + 8x²} is a basis for P₂. Find the coordinates of p(x) - [p(x)]B = (56 +30x + 64x²) relative to this basis: