Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) R2 has infinitely many non-zero, proper vector subspaces.
ii) If T : V → W is a one-one linear transformation between two finite dimensional vector spaces V andW then T is invertible.
iii) If Ak = 0 for a square matrix A, then all the eigenvalues of A are zero.
iv) Every unitary operator is invertible.
v) Every system of homogeneous linear equations has a non-zero solution.
Let's go through each statement and determine whether it is true or false, providing a justification for each:
i) R^2 has infinitely many non-zero, proper vector subspaces.
TRUE.
Justification: In R^2, you can find infinitely many non-zero, proper vector subspaces. For example, you can take any line passing through the origin in R^2, and it will be a proper vector subspace. There are infinitely many such lines, so there are infinitely many non-zero, proper vector subspaces.
ii) If T: V → W is a one-one (injective) linear transformation between two finite-dimensional vector spaces V and W, then T is invertible.
TRUE.
Justification: If T is injective, it means that for every element in V, there is a unique element in
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Let V = R3 andW = {(x,y, z)|x+y+z = 0} be a subspace of V. Which of the
following pairs of vectors are in the same coset of W in V?
i) (1,3,2) and (2,2,2). ii) (1,1,1) and (3,3,3).
Find the orthogonal canonical reduction of the quadratic form
x2+y2 + z2 - 2xy - 2xz - 2yz. Also, find its principal axes.
Find the inverse of the matrix
using row reduction.
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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Which of the following are binary operations on S = {x ∈ R | x > 0}? Justify your answer.
- The operation ∇ defined by x∇ y = |ln(xy)| where ln x is the natural logarithm.
- The operation Δ defined by x Δ y = x2 + y3.
Also, for those operations which are binary operations, check whether they are associative and commutative.
Complete the set S = {x3+x2+1,x2+x+1,x+1} to get a basis of P3
Check whether the following system of equations has a solution.
4x + 2y + 8z + 6w = 3
2x + 2y + 2z + 2w = 1
x + 3z + 2w = 3