4. Determine whether each of the following sets is a subspace of the vector space P4 (R) of polyno- mials of degree at most 4. (a) U = {fP4 (R) | f(-x) = f(x)} (b) V = {fe P4(R) | f(1) = 1}.

Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.

Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].

33. Let V = P3 be the vector space of polynomials of degree <3, and let p₁(x) = (2-x)² and p2(x) = (2x + 1)2. Determine if g(x) = x² + 1 is in Span(P₁, P2).

(3) For each of the following sets, determine whether it is linearly independent or dependent. (0) € 9000 3 (c) {x³x, 2x² + 4,-2x³ +3x²+2x+6} in P₂ (R), where P3 (R) denotes the vector space of polynomials over the reals of degree at most 3. -1 2 (d) {(-12 1) (1 7¹). (1¹3) (3¹2)} in M2x2(R), where M2x2(R) 0 denotes the vector space of 2 × matrices over the reals. {0·0)} span(S), i.e., orthogonal to every vector in span(S). (b) (4) Let S in R4. 7 in R4. CR4. Find all vectors in R4 that are orthogonal to (5) Let A and B be matrices over the reals. For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. (a) If A + B is defined, then rank(A + B) = rank(A) + rank(B). (b) If AB is defined, then rank(AB) = rank(A) rank(B). (c) If A has size m × n, then rank(A) ≤ min{m, n}. (d) rank(A) = rank(At) (e) nullity(A) = nullity(A¹)

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

,x43x-2 4. Determine whether the veltars X-I and X²-3x+2 dre linearly independent Or dependent in P2 (te vector Space of Polynomials of doyree less than equal to 2). or

Suppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 to show each possibility.

23. Let fi = 1+2x + 3x2, f2 = 2+3x + 4x². B = (f1, f2) is a basis for the subspace V = {ao + a1x + a2x²|ao – 2a1 + a2 = 0} of R2[x]. (:) Let g = 2+ 5x + 8x2. If [g]B = S then s = a) -5 b) -4 c) 3 d) 4 e) g is not in V.

Let P3 be the vector space of all polynomials of degree 3 or less in the variable z. Let = 2+x+x², 2+x+x², 2+x², = 11 + 3x + 6x² choose PI(T) P2(x) P3(x) = P4(x) and let C = {p1(x), p2(x), P3(x), P4(x)}. a. Use coordinate representations with respect to the basis B = {1, 2, ², ³} to determine whether the set C forms a basis for P.. = c. The dimension of span(C) is b. Find a basis for span(C). Enter a polynomial or a comma separated list of polynomials. {}