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.
We are asked to determine true or false for each statement and justify manually with short proofs or counterexamples. Let’s go one by one. --- ### (i) Statement: If $W_1$ and $W_2$ are proper subspaces of a non-zero finite-dimensional vector space $V$ and $$ \dim(W_1) > \frac{\dim(V)}{2}, \quad \dim(W_2) > \frac{\dim(V)}{2}, $$ then $W_1 \cap W_2 \neq \{0\}$. Proof: Let $\dim(V) = n$. Then $$ \dim(W_1) + \dim(W_2) > \frac{n}{2} + \frac{n}{2} = n $$ By the dimension formula for subspaces: $$ \dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2) $$ Since $\dim(W_1 + W_2) \le n$, we get $$ \dim(W_1 \cap W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 + W_2) \ge \dim(W_1) + \dim(W_2) - n > 0 $$ Hence $W_1 \cap W_2 _________ __________ _____ ___ ______ __________ __________ __________ _________ _____ ______.
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Check whetherr ? ∶ ℝ2 → ℝ2, defined by ? (?, ?) = (−?, ?) is a linear transformation.
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.
Check that is a basis for ?2 , the vector space of polynomials with real coefficients of degree ≤ 2.
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Let ? be any non-empty set and let ? (?) be the set of all real valued functions on ℝ. Define addition on and scalar multiplication by
Check that
is a vector space.
represent the coordinates with respect to the bases
find the representation of ? in terms of (?1 , ?2 , ?3).
Let ? ∶ ℝ3 → ℝ3 be a linear operator and suppose the matrix of the operator with respect to the ordered basis
. Find the matrix of the linear transformation with respect to the basis