Algebra and Geometry, Set 3, A.Sz.
Linear spaces
1. Is W a subspace of the linear space V over R?
1.1. V = R2 W = {(x, y) ∈ R2 : xy 6= 0} 1.3. V = R3
1.2. V = C
W = {z ∈ C : z 3 ∈ R}
W = {(x, y, z) ∈ R3 : y = 3x − z}
1.4. V = P(R) W = {p ∈ P(R) : p(π) = 0}
2. For what a, b ∈ R
2.1. is (1, 3, 2) a linear combination of (2, 3, 1) and (a, 8, 3)?
2.2. is 2x3 + 2x2 + 3 a linear combination of ax2 − 8x − 1 and x3 − bx2 + ax + 1?
3. Is S = {(1, 0, 1, 1), (1, 0, 0, 1), (1, 1, 2, 3)} linearly independent? Is S a basis of R3 ?
Does (4, −1, 0, 3) belong to Span S?
4. Is S linearly independent? Is S a basis of V over R?
4.1. V = R3
S = {(−1, 4, 5), (5, 0, 1), (2, 2, 3)}
4.2. V = C
S = {1, i}
4.3. V = P3 (R)
S = {x + 2, (x + 2)2 , (x + 2)3 }
4.4. V = P2 (R)
S = {x − 3, 4x, 3x2 − x + 5}
4.5. V = R3
S = {(1, 0, 2), (0, 1, 1), (3, 1, 4), (4, 5, 2)}
5. Find a basis and the dimension of V over R.
5.1. V = Span{(2, 0, 1), (3, 1, 0), (1, 3, −4)}
5.2. V = {(x, y, z) ∈ R3 : 7x − 2y + z = 0}
5.3. V = {p ∈ P3 (R) : p(−1) = p(0)}
Practise
1. Find a basis and the dimension of V = {(x1 , x2 , x3 , x4 ) ∈ R4 : 7x1 +x2 = 0∧2x1 +5x3 −x4 = 0}.
2. Is V = {(x, y, z) ∈ R3 : |3x + 2y − z| = |y + 2z|} a subspace of R3 ?
3. Is S = {x3 + x + 2, 2x3 + x2 + 2x + 1, 3x3 + x + 1} linearly independent?
Is S a basis for P3 (R)? Does 4x3 − 3x2 + 9 belong to Span S?
4. Find a basis and the dimension of W = {z ∈ C : 5z̄ + 3z + 4iz = 0} over R.
5. Prove that S = {(1, −1, 2), (0, 2, 1), (1, 3, 0)} is a basis for R3 .
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