Find which of the axioms will be violated if addition of vectors in problem 3 is changed to A+B = A S B

Adv.

In the following, find out whether S froms a subspace of V ? (a) V = R3, S = {(x1, x2, x3) : x1 + 5x2 + 3x3 = 0} (b) V = R3, S = {(x1, x2, x3) : x1 + 5x2 + 3x3 = 1} (c) V = R3, S = {(x1, x2) : x1 ≥ 0, x2 ≥ 0} (d) V = P(R), the set of all polynomials over reals and S = {p(x) ∈ P(R) : P(5) = 0} (e) V = Rn, S = {(x1, x2, ..., xn) : x1 = x2}

Prove or disprove: (a) Union of two subspaces of V is a subspace of V . (b) Intersection of any number of subspaces is a subspace.

Check whether the following set of vectors are linearly dependent or independent. (a) S = {(1, 2, −2, −1),(2, 1, −1, 4),(−3, 0, 3, −2)} (b) S = {(1, 3, −2, 5, 4),(1, 4, 1, 3, 5),(1, 4, 2, 4, 3),(2, 7, −3, 6, 13)}

Whether the system below is consistent? Justify. x + 2y − 3z = 1 3x − y + 2z = 5 5x + 3y − 4z = 2

Solve x + 2y − 3z + 2w = 2 2x + 5y − 8z + 6w = 5 3x + 4y − 5z + 2w = 4

For the system x + 2y − z = 0 2x + 5y + 2z = 0 x + 4y + 7z = 0 x + 3y + 3z = 0

Find the solution space as well as dimension.

All natural numbers and 0 are called the ……………….. numbers?

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