Are the following statements true or false?+ 1. The set (0) forms a basis for the zero subspace.+ 2. If Si is of dimension 3 and is a subspace of R*, then there can not exist a subspaceS2 of Rª such that Si C S2 C Rª with S1 # S2 and S2# R*.3. Let m > n. Then U = {u], un. ...,Um) in R" can form a basis for R" if the correct m –n vectors are removed from U.+ 4. Let m < n. Then U = {u],u2 ... , Um in R" can form a basis for R" if the correct n – m vectors are added to U.+ 5. R" has exactly one subspace of dimension m for each of m = 0, 1,2, ... , n .

Since every subspace has a basis, zero subspace has a basis of dimension 0. I, e in the basis of…

Answered: Are the following statements true or…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let V=R2 and let H be the subset of V of all points on the line 3x - 2y=-6. Is H a subspace of the vector space V? 1. Is H nonempty? choose # 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H. using a comma separated list and syntax such as , . 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated list and syntax such as 2, . 4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. choose 7
Let W be the set of all vectors of the form shown on the right, where b and care arbitrary. Find vectors u and v such that W = Span{u, v}. Why does this show that W is a subspace of R³? Using the given vector space, write vectors u and v such that W = Span{u, v}. {u, v} = {} (Use a comma to separate answers as needed.) 7b + 6c -b 8c
linear algebra 3.1 Q1 a, c, e
Let {w1, w2, ..., Wk} be a basis for a subspace W of the vector space V. Let v be a vector in W1. Show that 1 (v, w1) + 2 (v, w2) + 3 (v, w3) + ...+ k (v, wk) cannot be a negative number.
check the image for question
Let fi = 1+2x + 3x², f2 = 2 + 3x + 4x². B = (f1, f2) is a basis for the subspace V = {ao +a1x+azx²[ao – 2a1 +a2 = 0} of R2[r]. Let g = 1+5x + 9x². If [g]g = (,) then s = a) -7 b) -5 c) 5 d) 7 e) g is not in V.
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2. If U and W are subspaces of V such that V = U+W and UnW = {0}, then prove that every vector in V has a unique representation of the form u +w, where u is in U and w is in W. V is called the direct sum of U and W, and is written as V=U @W. Direct sum Which of the sums in part (1) are direct sums? 3. Let V=UW, and let {u₁, 2,..., uk} be a basis for the subspace U and {w₁, W2,..., Wm} be a basis for the subspace W. Prove that the set {u₁,..., uk, w₁,..., Wm} is a basis for V. 4. Consider the subspaces U = {(x, 0, y): x, y ≤ R} and W = {(0, x, y): x, y = R} of V = R³. Show that
-3 1 -10 3 -2 2 -2 -3 and let W the subspace of R4 spanned by i and õ. Find a basis of w+, the orthogonal complement of W in R4. ||
Which of the following is a subspace of the vector space P2 of all polynomials of degree at most 2? Select one or more: O a. The set of all polynomials of degree at most 1 O b. {ao +a1r + a2x2 E P2 : ao = 4} Oc. {ao +a1r + a2x E P2: a1 + a2 = 3} %3D d. {1+x+ x}
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