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.

Adv.

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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