Describe the linear transformation ? ∶ ℝ2 → ℝ2 such that where ? is the standard basis of ℝ 2
We are given a linear transformation $$ T: \mathbb{R}^2 \to \mathbb{R}^2 $$ with matrix representation with respect to the standard basis $B = \{ \mathbf{e}_1, \mathbf{e}_2 \}$, where: $$ [T]_B = \begin{bmatrix} 1 & 2 \\ 2 & 0 \end{bmatrix} _________ __________ ___ ______ ____.
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represent the coordinates with respect to the bases
find the representation of ? in terms of (?1 , ?2 , ?3).
Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample
i) If ?1 and ?2 are proper subspaces of a non-zero, finite dimensional, vector space ? and
ii) If ? is a vector space and is such that ?? ≠ ?? if ? ≠ ?, then S
is a linearly independent set
iii) ? are linear operators on a finite dimensional vector space ? and ?1 ∘ ?2 is invertible, ?2 ∘ ?1 is also invertibl
iv) If an ? × ? square matrix, ? ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial
v) If ?1 , ?2 ∶ ? → ? are self adjoint operators on a finite dimensional inner product space ?, then ?1 + ?2 is also a self adjoint operator.
Verify Cayley-Hamilton theorem for the matrix
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
Find the kernel of the linear transformation ? ∶ ℝ2 → ℝ2 defined by
Check that is a basis for ?2 , the vector space of polynomials with real coefficients of degree ≤ 2.
Solve the folowing set of simultaneous equations using Cramer’s rule:
3? + ? + ? = 0