Check whether each of the following subsets of R3 is linearly independent.
i) {(1,2,3), (-1,1,2), (2,1,1)}. ii) {(3,1,2); (-1,-1,-3), (-4,-3-0)}.
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Check whether the vector (2,sqrt(3),3) sis equally inclined to the vectors (2,2 sqrt(3)) and (4,0).
Find the inverse of the matrix
using row reduction.
Check whether the following system of equations has a solution.
4x + 2y + 8z + 6w = 3
2x + 2y + 2z + 2w = 1
x + 3z + 2w = 3
Let T : P2 → P1 be defined by
T (a + bx + cx2) = b + 2c + (a − b)x.
Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases B1 = {x2, x2 + x, x2 + x + 1} and B2 = {1, x}. Find the kernel of T. .
Let P(e) (x) = { p(x) ∈ R[x]| p(x) = p(−x)}
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| Find W = P(e) ∩ P3. Find a basis for W and find the dimension of W . |
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Define T : R3 → R3 by
T (x, y, x) = (x + y, y, 2x − 2y + 2z).
Check that T satisfies the polynomial (x − 1)2(x − 2). Find the minimal polynomial of T .
Consider the basis e1 = (−2, 4, −1), e2 = (−1, 3, −1) and e3 = (1, −2, 1) of R3
over R. Find the dual basis of {e1, e2, e3}.