Q1 4 Points In each part, determine if the given set H is a subspace of the given vector space V. Q1.1 1 Point V=R4 and H is the set of all vectors in R4 which have the form (e.g., the vector x = 園 2 a b x = for some scalars - 2a 1 +36 a, b. is an example of one element of H.) Q1.4 1 Point V=C(R), the vector space of all real-valued continuous functions whose domain is the set R of all real numbers, and H is the subset of C(R) consisting of all of the constant functions. (e.g. the function ƒ : R → R defined by the formula f(x) = 3 for all x E R is an example of one element of H.) OH is a subspace of V. H is not a subspace of V. Save Answer

Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}

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].

Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.

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.

Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.

Prove that in a given vector space V, the zero vector is unique.

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.

In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V= F, W=finF:f(0)=1

In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 37. V = P, W is the set of all polynomials of degree 3

Find a basis for R2 that includes the vector (2,2).

Let T be a linear transformation from R2 into R2 such that T(1,2)=(1,0) and T(1,1)=(0,1). Find T(2,0) and T(0,3).

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.