2. Let P3(R) be the vector space of all polynomials of degree 3 or less (and the zero polynomial). Determine whether p(x) = x³ – 5x + 8 is in the subspace (S), where S = {67x* + x² + 9x + 4, 10x³ – 50x² + x + 6, 7x* + 23x² – x + 17}.

b) Is W = {f(x):f is a polynomial with degree <3,f (0) 0} a subspace of P: all 3 ? Explain your answer. polynomials with degree

Let P₂ be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 22x² — 29x – 18, 11x² – 14x – 9 and 27x - 22x² + 18. a. The dimension of the subspace His b. Is {22x² 29x - 18, 11x² - 14x · 9, 27x − 22x² + 18} a basis for P₂? choose c. A basis for the subspace H is { Be sure you can explain and justify your answer. Enter a polynomial or a comma separated list of polynomials, where you can enter xx in place of ².

6. P2 denotes the set of all polynomials of degree ≤ 2, in a single variable x, with real number coefficients. P2 is a vector space under the usual operations of polynomial addition (combining like terms) and scalar multiplication. Determine whether each set of polynomials in P2 is linearly independent or linearly dependent. If the set is linearly dependent, find a dependence relation. If the set is linearly independent, demonstrate that any dependence relation has to be trivial. (a) {2x, x+3} (b) {x, 2x+5, 10} (c) {x² + 5x+1, x²-7x, x² +10, 2x² + 3x-4} 1ST (d) Make a linearly independent set of three polynomials in P2. Explain how you know.

Is the set of noninvertible 2 × 2 matrices a subspace of R2×2 ?

? Solve

Let S be the set of all real 2x2 matrices whose rows are orthogonal vectors. Show whether S is a subspace of Mat2x2 (R)

5. Let P2 be the vector space of polynomials of degree at most 2. Determine if the vector v = 10x – 7 is in the subspace Span{3x? + x – 1, x² – 3x + 1}

Find the dimension of the subspace Hof R2 spanned by

one example of a 4-dimensional vector space that is not R 4