represent the coordinates with respect to the bases
find the representation of ? in terms of (?1 , ?2 , ?3).
We are asked to represent the quadratic form $$ Q(X) = x_1^2 - 4x_1x_2 + 2x_2x_3 + x_2^2 + x_3^2 $$ in terms of the coordinates $(y_1, y_2, y_3)$, where: $$ B_1 = \big( (1,0,0), (1,1,0), (0,0,1) \big), \quad B_2 = \big( (1,0,0), (0,1,1), (0,0,1) \big) $$ Here $(x_1, x_2, x_3)$ are the coordinates of a vector in basis $B_1$, and $(y_1, y_2, y_3)$ are coordinates in basis $B_2$. We need to change the coordinates from $x$-basis to $y$-basis and then rewrite $Q$. ______ ___ _________ ___ _________ ___.
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Consider the linear operator ? ∶ ℂ3 → ℂ3 , defined by
i) Compute ? ∗ and check whether ? is self-adjoint
ii) Check whether ? is unitary.
Find the matrix of the linear transformation with respect to the ordered basis{(0, −1), (−1, 0)}.
Let ? be a 2 × 3 matrix, ? be a 3 × 4 matrix and ? be a 3 × 2 matrix and ? be a 3 × 4 matrix. Is ?? + ??? defined? Justify your answer.
Let ? be the vector space of all real valued functions that are twice differentiable in ℝ and
Check that ? is a linearly independent set over ℝ. (Hint: Consider the equation
Check whether is a subspace of ℝ3
Verify Cayley-Hamilton theorem for the matrix
Find the vector equation of the plane determined by the points (1, 0, −1), (0, 1, 1) and (−1, 1, 0)