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About 11892 results. 1294 free access solutions
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| m95808 | Let V = C [a, b] be the vector space of all real-valued functions that are integrable over the interval [a, b]. Let W = R1 Define L: V → W by L(f) = ∫ab f(x) dx. Prove that L is a linear transformation. |
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| m95809 | Let v denote a vector in an inner product space V.
(a) Show that W = (w | w in V,(v, w) = 0} is a subspace of V.
(b) If V= R3 with the dot product, and if v = (1, -1, 2), find a basis for IV. |
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| m95810 | Let V = M22 and let L : V → V be the linear operator defined by L(A) = AT, for A in V. Let S = {A1, A2, A3, A4}, where
(a) Find [L(Ai)]s for i = 1,2,3,4.
(b) Find the matrix B representing L with respect to 5.
(c) Find the eigenvalues and associated eigenvectors of B.
(d) Find the eigenvalues and associated eigenvectors of L? |
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| m95811 | Let v, w be two vertices in Kn, n ≥ 3. How many walks of length 3 are there from v to w? |
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| m95812 | Let V = {x1, x2, . . . , xv] be the set of varieties and {B1, B2, . . ., Bb] the collection of blocks for a (v, b, r, k, λ)- design. We define the incidence matrix A for the design by <br><div align="center"><img src="../image/images12/954-M-L-A-L-S (8666)-1.png"/></div>a) How many l s are there in each row and column of A? <br>b) Let Jm×n be the m × n matrix where every entry is 1. For Jnxn we write Jn. Prove that for the incidence matrix A, A ∙ Jb = r J y × b and Jv ∙ A = k J v × b <br>(c) Show that <br><div align="center"><img src="../image/images12/954-M-L-A-L-S (8666)-2.png"/></div>Where Iv is the v × v (multiplicative) identity, <br>(d) Prove that <br>det(A ∙ An) = (r - λ)n-1[r + (v - 1)λ] = (r - λ)v-lrk. <br> </span> </div> </div> <!-- Google Adword Banner--> <div class="google-adword-banner"> <div class="adword-banner"> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script> <!-- question_page --> <ins class="adsbygoogle" style="display:inline-block;width:730px;height:90px" data-ad-client="ca-pub-4274030376980924" data-ad-slot="7315349492"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> </div> <!-- Relevant Question Section--> <div class="related-question-section"> <span class="related-question-section-heading">Students also viewed these questions</span> |
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| m95813 | Let V1, V2, and V3 be vector spaces of dimensions n, m, and p, respectively. Also let L1: V1 → V2 and L2: V2 → V3 be linear transformations. Prove that L2 o L1: V1 → V3 is a linear transformation. |
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| m95815 | Let Vbc an w-dimensional inner product space, and let T and S denote symmetric linear operators on V. Show that:
(a) The identity operator is symmetric.
(b) rT is symmetric for all r in R.
(c) S + T is symmetric.
(d) If T invertible, then T-1 is symmetric.
(e) If ST = TS, then ST is symmetric. |
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| m95817 | Let W and U be subspaces of vector space V.
(a) Show that W ( U, the set of all vectors v that are either in W or in U, is not always a subspace of V.
(b) When is W ( U a subspace of V?
(c) Show that W ( U, the set of all vectors v that are in both W and U, is a subspace of V. |
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| m95818 | Let W be a nonempty subset of a vector space V. Prove that W is a subspace of V if and only if ru + sv is in W for any vectors u and v in W and any scalars r and s. |
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| m95819 | Let W be a subspace of an inner product space V and let {w1, w2, ... wm} be an orthogonal basis for W. Show that if v is any vector in V, then |
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| m95821 | Let W be an n × 1 matrix such that WT W = 1. The n × n matrix
H = In - 2WWT
is called a Householder* matrix.
(a) Show that H is symmetric.
(b) Show that H-1 = HT. |
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| m95822 | Let w be the complex number (1/√2) (1 + i).
(a) Show that w8 = 1 but of wn ≠ 1 for n ∈ Z+, 1 ≤ n ≤ 7.
(b) Verify that [wn|n ∈ Z+, 1 ≤ n ≤ 8] is an abelian group under multiplication. |
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| m95823 | Let W be the plane R3 given by the equation
In Exercise, find projWv for the given vector v and subspace W.
(a)
(b)
(c) |
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| m95824 | Let W be the sample space for an experiment and let A, B, C ⊂ . If events A, B are independent, events A, C are disjoint, and events B, C are independent, find Pr(B) if Pr(A) = 0.2, Pr(C) = 0.4, and Pr(A U B U C) = 0.8. |
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| m95825 | Let W be the set of all 2 ( 2 matrices
Such that Az = 0, where
Is W a subspace of M22? Explain. |
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| m95826 | Let W be the set of all 3 ( 3 matrices of the form
Show that W is a subspace of M33. |
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| m95827 | Let W be the subspace of the Euclidean space R3 with basis
Let
(a) Find the length of v directly
(b) Using the Gram-Schmidt process, transform S into an orthonormal basis T for W. |
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| m95828 | Let ∑ = {w, x, y] and ∑2 = {x, y, z} be alphabets. If A1 = {xtyj|i, j ∈ Z+, j > i > 1}, A2 = {wtyJi, j ∈ Z+, i > j > 1}, A3 = {wlxJylzJi, j ∈ Z+, j > i > 1}, and A4 = {zJ (wz)lwJi, j ∈ Z+, i > 1, j > 2}, determine whether each of the following statements is true or false۔
(a) Ai is a language over ∑1.
(b) A2 is a language over ∑2.
(c) A3 is a language over ∑1 ∪ ∑2.
(d) A] is a language over ∑1 ∩ ∑2.
(e) A4 is a language over ∑1 △ ∑2.
(f) A] U A2 is a language over ∑1. |
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| m95830 | Let W1 and W2 be subspaces of a vector space V. Let W1 + W2 be the set of all vectors v in V such that v = w1 + w2, where w1 is in W1 and w2 is in W2. Show that W1 + W2 is a subspace of V. |
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| m95831 | Let W1 and W2 be subspaces of a vector space V with W1 ( W2 = {0}. Let W1 + W2 be as defined in Exercise 34. Suppose that V = W1 + W2. Prove that every vector in V can be uniquely written as w1 + w2, where w1 is in W1 and w2 is in W2. In this case we write V = W1 W2 and say that V is the direct sum of the subspaces W1 and W2. |
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