If matrix A represents the orthogonal projection onto a plane V in R3, what is the

Which of the subsets of Pi given in Exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces.

Which of the subsets of Pi given in Exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces.

Which of the subsets of Pi given in Exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces. {pity P'(0 = Pi2)) iP 'is the derivative.)

Which of the subsets of Pi given in Exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces. {p(0: fo P(0 dt = l

Which of the subsets of Pi given in Exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces. {p(tY p(~t) = for all t}

Which of the subsets of M3x3 given in Exercises 6 through 11 are subspaces o/K3x3? The invertible 3 x 3 matrices

Which of the subsets of M3x3 given in Exercises 6 through 11 are subspaces o/K3x3?The diagonal 3 x 3 matrices

Which of the subsets of M3x3 given in Exercises 6 through 11 are subspaces o/K3x3? The upper triangular 3 x 3 matrices

Which of the subsets of M3x3 given in Exercises 6 through 11 are subspaces o/K3x3?The 3 x 3 matrices whose entries are all greater than or equal to zero

Which of the subsets of M3x3 given in Exercises 6 through 11 are subspaces o/K3x3?The 3 x 3 matrices A such that vector is in the kernel of A

Which of the subsets of M3x3 given in Exercises 6 through 11 are subspaces o/K3x3? The 3 x 3 matrices in reduced row-echelon form

Let V be the space of all infinite sequences of real numbers. (See Example 5.) Which of the subsets of V given in Exercises 12 through 15 are subspaces ofV? The arithmetic sequences [i.e., sequences of the form (a, a + k, a + 2ky a + 3k,...), for some constants a and k]

Let V be the space of all infinite sequences of real numbers. (See Example 5.) Which of the subsets of V given in Exercises 12 through 15 are subspaces ofV?The geometric sequences [i.e., sequences of the form (a, ary ar2, <?r3, ...), for some constants a and r]

Let V be the space of all infinite sequences of real numbers. (See Example 5.) Which of the subsets of V given in Exercises 12 through 15 are subspaces ofV? The sequences ( jc o, x\,...) that converge to zero (i.e., lim xn = 0) n-> 00

Let V be the space of all infinite sequences of real numbers. (See Example 5.) Which of the subsets of V given in Exercises 12 through 15 are subspaces ofV?Thesquare-summablesequences (*0, x\,...) (i.e., those 00 for which ^ xf converges)

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The real linear space C2

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension. The space of all matrices in R2x2 such A that a + d = 0

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension. The space of all diagonal 2 x 2 matrices

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all diagonal nxn matrices

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all lower triangular 2 x 2 matrices

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all upper triangular 3 x 3 matrices

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension. The space of all polynomials /(r) in P 2 such that /(I) = 0

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all polynomials f(t) in P3 such that /(l) = 0 and J\f(t)dt= 0

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all 2 x 2 matrices A that commute with 1 O' B = 0 2

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all 2 x 2 matrices A that commute with 1 f B = 0 1

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all 2 x 2 matrices A such that A '1 f o o' 1 1 0 0

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension. The space of all 2 x 2 matrices A such that 'l 2 A = 0 o' 3 6 0 0

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all 2 x 2 matrices S such that

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all 2 x 2 matrices S such that

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all 2 x 2 matrices 5 such that

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all 2 x 2 matrices 5 such that

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all 3 x 3 matrices A that commute with

Find a basis for each of the spaces in Exercises 16 through 9 and determine its dimension.The space of all 3 x 3 matrices A that commute with

If B is a diagonal 3 x 3 matrix, what are the possible dimensions of the space of all 3 x 3 matrices A that commute with B? Use Exercises 35 and 36 as a guide.

If B is a diagonal 4 x 4 matrix, what are the possible dimensions of the space of all 4 x 4 matrices A that commute with B1

What is the dimension of the space of all upper triangular nxn matrices?

If u is any vector in R", what are the possible dimensions of the space of all n x n matrices A such that Av = 0?

If B is any 3x3 matrix, what are the possible dimensions of the space of all 3 x 3 matrices A such that BA = 0?

If B is any nxn matrix, what are the possible dimensions of the space of all n x n matrices A such that BA = 01

If matrix A represents the reflection about a line L in R2, what is the dimension of the space of all matrices S such that AS = S 1 0 0 -1 (Hint: Write S = [5 >], and show that 5 must be parallel to L, and w must be perpendicular to L.)

If matrix A represents the orthogonal projection onto a plane V in R3, what is the dimension of the space of all matrices S such that AS = S 1 0 1 0 0 0 0 0 0 See Exercise 43.

Find a basis of the space V of ail 3 x 3 matrices A that commute with B = 0 1 0 0 0 0 and thus determine the dimension of V.

In the linear space of infinite sequences, consider the subspace W of arithmetic sequences (see Exercise 12).Find a basis for W, and thus determine the dimension of W.

A function /(/) from R to R is called even if f(-t) = /(r), for all t in R, and odd if / ( t) = - /(r), for all t. Are the even functions a subspace of F(R, R), the space of all functions from R to R? What about the odd functions? Justify your answers carefully.

Find a basis of each of the following linear spaces, and thus determine their dimensions. (See Exercise 47.) a. {/ in P 4: f is even) b. {/ in P 4: f is odd}

Let L(Rm, R) be the set of all linear transformations from Rm to Rn. Is L (Rm, R") a subspace of F(Rm, Rw), the space of all functions from Rm to R'1? Justify your answer carefully.

Find all the solutions of the differential equation f"(x) + 8 / ' ( jc) - 20f(x) = 0.

Find all the solutions of the differential equation / " ( * ) - 7 /'(jc)+ 1 2 /( jc) = 0.

Make up a second-order linear DE whose solution space is spanned by the functions e~x and e~5x.

Show that in an n-dimensional linear space we can find at most n linearly independent elements. Hint: Consider the proof of Theorem 4.1.5.

Show that if W is a subspace of an n-dimensional linear space V, then W is finite-dimensional as well, and dim(V^) < n. Compare with Exercise 3.2.38a.

Show that the space F(R, R) of all functions from R to R is infinite dimensional.

Show that the space of infinite sequences of real numbers is infinite dimensional.

We say that a linear space V is finitely generated if it can be spanned by finitely many elements. Show that a finitely generated space is in fact finite dimensional (and vice versa, of course). Furthermore, if the elements g l........gm span V, then dim(V) < m.

In this exercise we will show that the functions cos(*) and sin(jc) span the solution space V of the differential equation f"(x) = f(x). (See Example 1 of this section.) a. Show that if #(jr) is in V, then the function (g{x)) + (s'C*))2 is constant. Hint: Consider the derivative. b. Show that if g(;t) is in V, with #(0) = #'(0) = 0, then g(jt) = 0 for all x. c. If / (jc) is in V, then g(jc) = f(x) f (0) cos(x) / '( O)sin(jc) is in V as well (why?). Verify that (0) = 0 and #'(0) = 0. We can conclude that g(x) = 0 for all jc, so that /(jc) = / ( O)cos(x) + /'(0) sin(jc). It follows that the functions cos(x) and sin(jc) span V, as claimed.