Check whether is an eigenvector for the matrix
What is the corresponding eigenvalue?
To determine if the vector \(\begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}\) is an eigenvector of the matrix \[ A = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \end{bmatrix}, \] we need to check if there exists a scalar \(\lambda\) (the eigenvalue) such ___ ________ ________ _______ ___ __________ _______.
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Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) If W₁ and W₂ are proper subspaces of a non-zero, finite dimensional, vector space V and dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W₁ ∩ W₂ ≠ {0}.
ii) If Vis a vector space and S = {U1, U2,..., Un} CV, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set.
iii) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T1 ∘ T2 is invertible, T2 ∘ T₁ is also invertible.
iv) If an n x n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial.
If T1, T2: V → Vare self adjoint operators on a finite dimensional inner product space V, then T₁ + T₂ is also a self adjoint operator.
जाँच कीजिये कि B = {1, 2? + 1, (? − 1)2 }, ?2 के लिये एक आधार है जहाँ पी2 अधिक से अधिक कोटि P2 वाले वास्तविक गुणाँक बहुपदों की सदिश समष्टि है।
दिखाइये कि की एक उपसमष्टि है।
की उपस्मष्टि U का आधार भी ज्ञात कीजिये, जो
, को संतुष्ट करती है।
मान लीजिये कि A एक 2 x 3 आव्युह है, B एक 3 x 4 आव्यूह है, C एक 3 x 2 आव्यूह है और D एक 3 x 4 आव्यूह है। क्या AB + CtD परिभाषित है? अपने उत्तर की पुष्टि कीजिये।
Find the kernel of the linear transformation T: R2 → R2 defined by T(x, y) = (2x + 3y, 2x – 3y).
Let (x1, x2, x3) and (y1, y2, y3) represent the coordinates with respect to the bases B₁ = {(1,0,0), (1, 1, 0), (0, 0, 1)}, B₂ = {(1,0,0), (0, 1, 1), (0, 0, 1)}. If find the representation of Q in terms of (y1, y2, y3).
रैखिक रूपांतरण के क्रमित आधार {(0, −1), (−1, 0)} के सापेक्ष आव्यूह निकालिये, जो ? (?, ?) = (2?, ? − ?) द्वारा परिभाषित है,।.