please help me Q1-Q3, for Q1 why s can be solved but it said it is not closed un

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please help me Q1-Q3, for Q1 why s can be solved but it said it is not closed under scalar multiplication? Also, why Q2 the solution it write V1=1V1+ 0V2..... Please be able to follow up the comments

PRACTICE PROBLEMS 1. Show that the set H of all points in R2 of the form (3 s, 2+5 s) is not a vector space, by showing that it is not closed under scalar multiplication. (Find a specific vector u in H and a scalar c such that cu is not in H.) Hint: First write an equation that shows that vi s in 2. Let w = Span vi, , vp , where vi, , vp are in a vector space V. Show that vk is in W or l s W. Then adjust your notation for the general case.] p 3. An nxn matrix A is said to be symmetric ifAT -A. Let S be the set of all 3 x3 symmetric matrices. Show that S is a subspace of M3x3, the vector space of3x3 matrices SOLUTIONS TO PRACTICE PROBLEMS 1. Take any u inH say, u - d take any c1-say, c = 2. Then cu =|14|,Ifthis is in H, then there is some s such that 2+5s]-[14 That is, s = 2 and s = 12 / 5, which is impossible. So 2 u is not in H and H is not a vector space. 2. Vi vi+0+0Vp. This expresses vi as a linear combination ofv..., Vp. So Vi is in W. In general, vz is in Wbecaus,e 3. The subset S is a subspace of M3x3 since it satisfies all three of the requirements listed in the definition of a subspace: a. Observe that the 0 in M33 is the 3x3 zero matrix and since 00the matrix 0 is symmetric and hence 0 is in S. b. Let A and B in S. Notice that A and B are 3x3 symmetric matrices so AT -A and BT- B. By the properties of transposes of matrices, (A+BY=AT + Br = A+ B. Thus A + B is symmetric and hence A + Bis ins.

Explanation / Answer

1.

Set s=1 to get one vector in this set

u=(3,2+5)=(3,7)

Now multiplying this vector by scalar, c=3 gives

3(3,7)=(9,21)

Let, (9,21)=(3t,2+5t)

3t=9 gives ,t=3

2+5t=21 gives

t=19/5 hence no solution

So 3u is not in H and hence not a vectror space

2.

A vector,w, is in W if there exist a_1,..,a_p so that

w=a_1v_1+...+a_pv_p

WE se, a_k=1 and a_i=0 for al i not equal to k

SO we have w=v_k=0*v_1+...+1*v_k+...

Hence, v_k is in W

3.

Let, A,B be symmetric matrices and c a scalar

(A+B)^T=A^T+B^T=A+B

Hence, A+B is symmetric

Let, c be a scalar

(cA+cB)^T=(cA)^T+cB^T=c(A^T+B^T)=c(A+B)^T

Hence, closed under addition and scalar multiplication and hence a vector addition

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