10.Suppose that each of the vectors x(1), …, x(m) has n components, where n < m. Show that x(1), …, x(m) are linearly dependent. In each of Problems 11 and 12, determine whether the members of the given set of vectors are linearly independent for −∞ < t < ∞ . If they are linearly dependent, find the linear relation among them.

Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

1. Decide if p lies in the span of {v1, v2, v3}. If it does, find a linear combination that makes the vector. If it does not, show that no linear combination exists. p(x) = x − x³, v₁(x) = x², v2(x) = 2x + x², v3(x) = x+x³.

Show that the set S= {(20, 34), (40, 68)} is linearly independent or not?

Suppose S = {r,u, d} is a set of linearly independent vectors. If x = 2r + 2u + 2d, determine whether T = {r,u, x} is a linearly independent set. %3| v Select an Answer 1. Is T linearly independent or dependent? Linearly Independent Linearly Dependent n-trivial linear relation below. Otherwise, enter O's for the coefficients.

Q1. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, = (0,1,2) and x, =(-3,7,2) are linearly dependent and find the relation between them. Ans: 9x, - 12x, + 5х, - 5х, -0 Q2. If the vectors (0,1,a), (1, a,1) and (a,1,0) is linearly dependent, then find the value of a. Ans: 0,+/2 Q3. Find the eigen values and eigen vectors of the following matrices: 8 -6 2 (i) -6 7 [31 4] (ii) 0 2 6 0 o 5 -4 -4 3 Ans: (i) 0, 3, 15, k 2, ka (ii) 3, 2, 5, Q4. Verify Cayley-Hamilton theorem for the following matrix and hence compute A: [2 -1 1] A = -1 2 -1 I -1 2 [3 1 -1 Ans: 41 3 1 3 [2 11 Q5. Find the characteristic equation of the matrix A =0 1 0 and hence, compute A. Also find the matrix represented by A -5A" +7A“ - 3A +A* - SA' + 8A? - 2A +1. [8 5 5 [ 2 -1 -1 Ans: 2'-5a + 72 - 3 = 0, 0 3 0,A 3 55 8 3 -1 -1 10 5 Q6. Show that the matrix -2 -3 -4 has less than three linearly independent eigen vectors. Also 3 5 7 find them. Ans: A= 2,2,3. For i = 3, X, = [k,k,-2k] , for 2 = 2, X, = [5k,2k,-Sk] [i -1 2…

can u solve g,h,i

--16] -14|, and C 4] [47 5, B: [2] Let A 2 -11 Are A,B and C linearly dependent, or are they linearly independent? O Linearly independent O Linearly dependent If they are linearly dependent, determine a non-trivial linear relation. Otherwise, if the vectors are linearly independent, enter O's for the coefficients, since that relationship always holds. A+ B+ C = 0.

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