Linear Algebra 4th Edition solutions

Linear Algebra 4
Give three different bases for F2 and for M2X2(F)

Linear Algebra 4
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

Linear Algebra 4
In each part, determine whether the given vector is in the span of S. (a) (2,-1,1), S = {(1,0,2), (-1,1,1)} (b) (-1,2,1), S = {(1,0,2), (-1,1,1)} (c) (-1,1,1,2), S = {(1,0,1,-1), (0,1,1,1)} (d) (2,-1,1,-3), S = {(1,0,1,-1), (0,1,1,1)} (e) -x 3 4-2x2 4-3x 4-3, S = {x3 4- x 2 4-x + l,x2 4- x 4- 1,X 4-

Linear Algebra 4
What arc the coordinates of the vector 0 in the Euclidean plane that satisfies property 3 on page 3? Justify your answer.

Linear Algebra 4
Give three different bases for F2 and for M2X2(F).

Linear Algebra 4
In Fn , let ej denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Prove that {ei,e 2 ,... ,e n } is linearly independent.

Linear Algebra 4
Show that Pn(F) is generated by {1, x,... , x n }

Linear Algebra 4
Determine whether the following sets are subspaces of R3 under the operations of addition and scalar multiplication defined on R3 . Justify your answers. (a) W) = {(01,02,03) R3 : ai = 3a2 and 03 = a2) (b) W2 - {(01,02,03) R3 : a, = a 3 + 2} (c) W3 = {(01,02,03) R3 : 2ai - 7o2 + a 3 = 0} (d) W4 =

Linear Algebra 4
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

Linear Algebra 4
For each list of polynomials in P3(K), determine whether the first polynomial can be expressed as a linear combination of the other two. (a) (b) (c) (d) (e) (f) x 3 - 3x 4- 5, x3 4- 2x2 - x + 1, x3 + 3x2 - 1 4x3 + 2x2 - 6, x3 - 2x2 + 4x + 1,3x3 - 6x2 + x + 4 -2x3 - 1 lx2 4- 3x 4- 2, x3 - 2x2 + 3x - 1

Linear Algebra 4
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.

Linear Algebra 4
Label the following statements as true or false. (a) The zero vector is a linear combination of any nonempty set of vectors. (b) The span of 0 is 0 . (c) If 5 is a subset of a vector space V, then span(5) equals the intersection of all subspaces of V that contain S. (d) In solving a system of linear

Linear Algebra 4
Solve the following systems of linear equations by the method introduced in this section. (a) (b) (c) (d) (e) (f) 2si - 1x2 - 3x3 = - 2 3xi 3x2 2x3 + 5x4 = 7 X\ X2 2x3 x4 = 3 3xi - lx2 + 4x3 = 10 x\ - 2x2 + X3 = 3 2x\ X2 2x3 = 6 x\ + 2x2 X3 + a*4 = 5 xi 4- 4x2 - 3x3 3x4 6 2xi 4- 3x2 X3 4-

Linear Algebra 4
In each pari, apply the Gram Schmidt process to the given subset S of the inner product space V to obtain an orthogonal basis for span(S'). Then normalize the vectors in this basis to obtain an orthonormal basis ri for span(S'). and compute the Fourier coefficients of the given vector relative to 0.

Linear Algebra 4
For each of the following linear operators T on a vector space V, test T for diagonalizability. and if T is diagonalizable, find a basis 0 for V such that [T]a is a diagonal matrix. (a) V = P3(R) and T is defined by T(/(x)) = f'(x) + f"(x), respectively. (b) V = P2(R) and T is defined by T(ax2 + bx +

Linear Algebra 4
Let V denote the set of all differentiablc real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3.

Linear Algebra 4
Recall from Example 3 in Section 1.3 that the set of diagonal matrices in M2x2(^) is a subspace. Find a linearly independent set that generates this subspace.

Linear Algebra 4
For Exercises 2 through 6, prove that T is a linear transformation, and find bases for both N(T) and R(T). Then compute the nullity and rank of T, and verify the dimension theorem. Finally, use the appropriate theorems in this section to determine whether T is one-to-one or onto.

Linear Algebra 4
Show that the vectors (1,1,0), (1,0,1), and (0,1,1) generate F3 .

Linear Algebra 4
Write the zero vector of 3. If "3x4 (F).