Label the following statements as true or false. (a) The zero vector space has no basis

Label the following statements as true or false. (a) The zero vector space has no basis. (b) Every vector space that is generated by a finite set has a basis. (c) Every vector space has a finite basis. (d) A vector space cannot have more than one basis. 54 Chap. 1 Vector Spaces (e) (f) (g) (h) (i) (j) (k) (1) If a vector space has a finite basis, then the number of vectors in every basis is the same. The dimension of Pn (F) is n. The dimension of MmXn (F) is rn 4- n. Suppose that V is a finite-dimensional vector space, that Si is a linearly independent subset of V, and that S2 is a subset of V that generates V. Then Si cannot contain more vectors than S2. If S generates the vector space V, then every vector in V can be written as a linear combination of vectors in S in only one way. Every subspace of a finite-dimensional space is finite-dimensional. If V is a vector space having dimension n, then V has exactly one subspace with dimension 0 and exactly one subspace with dimension n. If V is a vector space having dimension n, and if S is a subset of V with n vectors, then S is linearly independent if and only if S spans V

Determine which of the following sets are bases for P2(R). (a) {-l- x + 2x2 ,2 + x-2x 2 ,l-2x4-4x 2 } ( b ) {14-2X4-X 2 , 3 4-X 2 ,X4-X 2 } (c) {1 - 2x - 2x2 , - 2 4- 3x - x 2 ,1 - x 4- 6x2 } (d) {- 1 4- 2x 4- 4x2 ,3 - 4x - 10x2 , - 2 - 5x - 6x2 } (e) {1 4- 2x - x 2 ,4 - 2x 4- x 2 , - 1 4- 18x - 9x2 }

Do the polynomials x 3 2x2 4T,4x2 x4-3, and 3x 2 generate P,3(i?)? Justify your answer

Is {(1,4, -6), (1,5,8), (2,1,1), (0,1,0)} a linearly independent subset of R 3 ? Justify your answer.

Give three different bases for F2 and for M2X2(F)

Give three different bases for F2 and for M2X2(F).

The vectors u : = (2,-3,1), u2 = (1,4,-2), u3 = (-8,12,-4), uA = (1,37, 17), and u*, = (3, 5,8) generate R3 . Find a subset of the set {ui,u2,U3,U4,ur,} that is a basis for R

Let W denote the subspace of R5 consisting of all the vectors having coordinates that sum to zero. The vectors wi = (2, -3,4, -5,2), u2 = (-6,9, -12,15, -6), u3 = (3, -2,7, -9,1), u4 = (2, -8,2, -2,6), u5 = (-1,1,2,1, -3), uQ = (0, -3, -18,9,12), u7 = (1,0, -2,3, -2), u8 = (2, -1,1, -9,7) generate W. Find a subset of the set {ui, u2,... ,Ug} that is a basis for W.

The vectors m = (1,1,1,1), u2 = (0,1,1,1), u3 = (0,0,1,1), and W4 = (0,0,0,1) form a basis for F4 . Find the unique representation of an arbitrary vector (01,02,03,04) in F4 as a linear combination of Mi, u2, U3, and U4

In each part, use the Lagrange interpolation formula to construct the polynomial of smallest degree whose graph contains the following points. (a) (-2,-6), (-1,5), (1,3) (b) (-4,24), (1,9), (3,3) (c) (-2,3), (-1,-6), (1,0), (3,-2) (d) (-3,-30), (-2,7), (0,15), (1,10)

Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is a basis for V, then {u 4- v 4- w, v + w, w} is also a basis for V.

Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is a basis for V, then {u 4- v 4- w, v + w, w} is also a basis for V.

The set of solutions to the system of linear equations xi - 2x2 4- x3 = 0 2xi 3x2 4- X3 = 0 is a subspace of R3 . Find a basis for this subspace.

Find bases for the following subspaces of F5 : Wi = {(ai,a2,a 3 ,a4,a 5 ) F5 : ax - a 3 - a4 = 0} and W2 = {(ai, a 2 , a3, a 4 , a 5 ) F5 : a2 = a 3 = a4 and ax 4- a 5 = 0}. What are the dimensions of Wi and W2?

The set of all n x n matrices having trace equal to zero is a subspace W of Mnxn (F) (see Exam is the dimension of W?

The set of all upper triangular n x n matrices is a subspace W of MnXn (F) (see Exercise 12 of Section 1.3). Find a basis for W. What is the dimension of W?

The set of all skew-symmetric n x n matrices is a subspace W of Mnxn(^) (see Exercise 28 of Section 1.3). Find a basis for W. What is the dimension of W?

Find a basis for the vector space in Example 5 of Section 1.2. Justify your answer

Complete the proof of Theorem 1.8.

Let V be a vector space having dimension n, and let S be a subset of V that generates V. (a) Prove that there is a subset of S that is a basis for V. (Be careful not to assume that S is finite.) (b) Prove that S contains at least n vectors.

Prove that a vector space is infinite-dimensional if and only if it contains an infinite linearly independent subset.

Let Wi and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and sufficient conditions on Wi and W2 so that dim(WinW2 ) = dim(W1).

Let Vi,V2, ,vk,v be vectors in a vector space V, and define Wi = span({vi,v2,...,vk}), and W2 = span({vi,v2,... ,vk,v}). (a) Find necessary and sufficient conditions on v such that dim(Wi) = dim(W2). (b) State and prove a relationship involving dim(Wi) and dim(W2) in the case that dim(Wi) ^ dim(W2)

Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x) Pn(R) there exist scalars Co, c\,...,cn such that g(x) = co/(x) 4- Cl /'(x) 4- c2f"(x) + 4- cnf^(x), where f^n '(x) denotes the nth derivative of f(x)

Let V, W, and Z be as in Exercise 21 of Section 1.2. If V and W are vector spaces over F of dimensions m and n, determine the dimension ofZ

For a fixed a R, determine the dimension of the subspace of Pn(R) defined by {f Pn(R): f(a) = 0}.

Let Wi and W2 be the subspaces of P(F) defined in Exercise 25 in Section 1.3. Determine the dimensions of the subspaces Wi n Pn (F) and W2 n Pn (F).

Let V be a finite-dimensional vector space over C with dimension n. Prove that if V is now regarded as a vector space over R, then dim V = 2n. (See Examples 11 and 12.)

Exercises 29-34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3.

Exercises 29-34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3.

Exercises 29-34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3.

Exercises 29-34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3.

Exercises 29-34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3.

Exercises 29-34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3.

Let W be a subspace of a finite-dimensional vector space V, and consider the basis {ui,u2,... ,uk} for W. Let {ui,u2,... ,uk,uk+\,... ,un} be an extension of this basis to a basis for V. (a) Prove that {uk+i + W, wfc+2 4- W,... , un 4- W} is a basis for V/W. (b) Derive a formula relating dim(V), dim(W), and dim(V/W).