Question: Let U be a nonempty subset of a vector space V. Show that U is a subspace of V if and only if u1 +

Let U be a nonempty subset of a vector space V. Show that U is a subspace of V if and only if u1 + au2 lies in U for all u1 and u2 in U and all a in R.

Step by Step Solution

★★★★★

3.39 Rating (152 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

If U is a subspace then u 1 au 2 is in U for any u i in U and a in R by the sub... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

950-M-L-A-L-S (6502).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

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.

Let U be a nonempty subset of Rn. Show that U is a subspace if and only if S2 and S3 hold.

Let S be a nonempty subset of a linear space, and let m = dim S = dim aff S. Suppose that x belongs to conv S so that there exist x1, x2, . . . , xn S and a1, a2, . . . , an R+ with a1,......,...

Let u and v be vectors in a vector space V. (a) Let H be any subspace of V that contains both u and v. Explain briefly why H also contains Span {u, v}. (b) Suppose {u, v} forms a basis for a subspace...

10. Determine whether each of the following statements is true or false and explain your answer. (a) If A is an m X n matrix and the equation AX = b is consistent for some b, then the columns of A...

questions: 2. (a) Show that {2:3 32,33 - I} is a basis for the subspace W = {a E RalIl = pl\") = all} = U}- of R3 [2:]. Extend this basis to a basis for Bali] and hence find a complement of W in R3...

Let W, Z be complementary subspaces of a vector space V, as in Exercise 2.2.24. Let V/W denote the quotient vector space, as defined in Exercise 2.2.29. Show that the map L:Z V/W that maps L[z] =...

for all question that has this letters r=3 s=2 t= 2 u = 1 v = 4 w = 0 1. (4 marks) For a = (r + s + t)/10 and b = (utv + w)/10 , let z = a + ib and w = -2 +i. sketch the complex plane and plot the...

Q1 part a to e parts..need solution in neat and clear handwriting on A4 page ASAP THEOREM 4.2.1 If W is a set of one or more vectors in a vector space V, then W is a subspace of V if and only if the...

Q2 part a to e parts..need solution in neat and clear handwriting on A4 page ASAP THEOREM 4.2.1 If W is a set of one or more vectors in a vector space V, then W is a subspace of V if and only if the...

Let A and B be compact subsets of a metric space (M. d). Show that both AU B and AnB are compact.

Identify the compound in each of the following pairs that reacts at the faster rate in an SN1 reaction: (a) Cyclopentyl iodide or 1-methylcyclopentyl iodide (b) Cyclopentyl bromide or...

Among others, the National Automobile Dealers Association website is a good source of price information for used cars. a . True b . False

YOU BE THE JUDGE WRITING PROB LEMConstruction Helicopters paid Heli-Dyne Systems $315,000 for three helicopters that were in Argentina. Two were ready to fly and one was disassembled for routine...

Let (a) Find an LU-decomposition of A. (b) Express A in the form A = L1DU1, where L1 is lower triangular with l's along the main diagonal, U1 is upper triangular, and D is a diagonal matrix. (c)...

Let (a) Prove: If a 0, then A has a unique LU-decomposition with l's along the main diagonal of L. (b) Find the LU-decomposition described in part (a). A =

Recall from Theorem 1.7.1 b that a product of lower triangular matrices is lower triangular. Use this fact to prove that the matrix L in 8 is lower triangular.

QUESTION 5 Monthly electric bills in NewJersey for a bellshaped distribution with a mean of $1 09 and a standard deviation of $37. a) About what percent of the bills will be between $72 and $146? b)...

Question 9 5 pts Cooper Tire has 45 million shares of common stock outstanding and 2 million bonds. The par value of the bonds is $1000. Cooper Tire's common stock is selling for $125 and its bonds...

Cam you please hand write. please i need the answers 13) A loan of $5,600 with an interest rate of 14% compounded annually can be repaid in 3 equal annual payments of $2,412.10. Complete the...