Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. (a) Let v₁ = x, V₁ = x² + 2, V3 list in R[x] 3. = 5. Prove that (V1, V2, V3) is a linearly independent [8] [6] (b) Let V1, V2, V3 be as defined in (a). Find a vector v₁ = R[x] 3 such that (V1, V2, V3, V4) is a basis of R[x] 3.

Let T be a linear transformation from R3 into R such that T(1,1,1)=1, T(1,1,0)=2 and T(1,0,0)=3. Find T(0,1,1)

Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.

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 V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.

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.

20. Let V denote the set of all real-valued functions f defined on the real line such that f(1) = 0. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3.

Let R[x] be a linear space with all polynomials p(x) = ax² + bx + c (a, b, c € R) whose degree is no more than 2. Let L be the operator on R[x]3 defined by 4 2 L(ax² + bx + c) = (a + 3b +6c)x² + (-=a 3b4c)x+ ²a + 3 3 (1) Find the matrix A representing L with respect to basis [1,2x, 3x²]; a + b + c) (2) Find the matrix B representing L with respect to basis [1-2x+3x2²,-1+2x, −1+3x²]; (3) Is the matrix A diagonalizable? Why?

From 1.21 and 1.22 linear algebra show all work

Q2. a) Let P be the set of all polynomials f(x), and let Q be the subset of P consistingof all polynomials f(x) so f(0) = f(1) = 0. Show that Q is a subspace of P.b) Let P be the set of all degree ≤ 4 polynomials in one variable x with real coefficients.Let Q be the subset of P consisting of all odd polynomials, i.e. all polynomials f(x) sof(−x) = −f(x). Show that Q is a subspace of P. Choose a basis for Q. Extend this basisfor Q to a basis for P. answer both qwestion according to linear algebra